Mereology

Mereology (from the Greek μερος,
‘part’)
is the theory of parthood relations: of the
relations of part to whole and the relations of part to part within a
whole.[1]
Its roots can be traced back to the early days of philosophy,
beginning with the Presocratics and continuing throughout the writings
of Plato (especially the Parmenides and the
Theaetetus), Aristotle (especially the Metaphysics,
but also the Physics, the Topics, and De
partibus animalium), and Boethius (especially De
Divisione and In Ciceronis Topica). Mereology occupies a
prominent role also in the writings of medieval ontologists and
scholastic philosophers such as Garland the Computist, Peter Abelard,
Thomas Aquinas, Raymond Lull, John Duns Scotus, Walter Burley, William
of Ockham, and Jean Buridan, as well as in Jungius's Logica
Hamburgensis (1638), Leibniz's Dissertatio de arte
combinatoria (1666) and Monadology (1714), and Kant's
early writings (the Gedanken of 1747 and the Monadologia
physica of 1756). As a formal theory of parthood relations,
however, mereology made its way into our times mainly through the work
of Franz Brentano and of his pupils, especially Husserl's third
Logical Investigation (1901). The latter may rightly be
considered the first attempt at a thorough formulation of a theory,
though in a format that makes it difficult to disentangle the analysis
of mereological concepts from that of other ontologically relevant
notions (such as the relation of ontological
dependence).[2]
It is not until
Leśniewski's Foundations of the General Theory of
Sets (1916) and his Foundations of Mathematics
(1927–1931) that a pure theory of part-relations was given an
exact formulation.[3]
And because Leśniewski's work was
largely inaccessible to non-speakers of Polish, it is only with the
publication of Leonard and Goodman's The Calculus of
Individuals (1940), partly under the influence of Whitehead, that
mereology has become a chapter of central interest for modern
ontologists and
metaphysicians.[4]

In the following we focus mostly on contemporary formulations of
mereology as they grew out of these recent
theories—Leśniewski's and Leonard and Goodman's. Indeed,
although such theories come in different logical guises, they are
sufficiently similar to be recognized as a common basis for most
subsequent developments. To properly assess the relative strengths and
weaknesses, however, it will be convenient to proceed in steps. First
we consider some core mereological notions and principles. Then we
proceed to an examination of the stronger theories that can be erected
on that basis.

A preliminary caveat is in order. It concerns the very notion of
‘part’ that mereology is about, which does not have an
exact counterpart in ordinary language. Broadly speaking, in English
we can use ‘part’ to indicate any portion of a given
entity. The portion may itself be attached to the remainder, as in
(1), or detached, as in (2); it may be cognitively or functionally
salient, as in (1)–(2), or arbitrarily demarcated, as in (3);
self-connected, as in (1)–(3), or disconnected, as in (4);
homogeneous or otherwise well-matched, as in (1)–(4), or
gerrymandered, as in (5); material, as in (1)–(5), or
immaterial, as in (6); extended, as in (1)–(6), or unextended,
as in (7); spatial, as in (1)–(7), or temporal, as in (8); and
so on.

(1)

The handle is part of the mug.

(2)

The remote control is part of the stereo system.

(3)

The left half is your part of the cake.

(4)

The cutlery is part of the tableware.

(5)

The contents of this bag is only part of what I bought.

(6)

That area is part of the living room.

(7)

The outermost points are part of the perimeter.

(8)

The first act was the best part of the play.

All of these uses illustrate the general notion of ‘part’
that forms the focus of mereology, regardless of any internal
distinctions. (For more examples and tentative taxonomies, see Winston
et al. 1987, Iris et al. 1988, Gerstl and Pribbenow
1995, Pribbenow 2002, Westerhoff 2004, and Simons 2013.) Sometimes, however, the
English word is used in a more restricted sense. For instance, it can
be used to designate only the cognitively salient relation of parthood
illustrated in (1), the relevant notion of salience being determined
by Gestalt factors (Rescher and Oppenheim 1955; Bower and Glass 1976;
Palmer 1977) or other perceptual and cognitive factors at large
(Tversky 2005). Or it may designate only the functional relation
reflected in the parts list included in the user's manual of a
machine, or of a ready-to-assemble product, as in (2), in which case
the parts of an object x are just its
“components”, i.e., those parts that are available as
individual units regardless of their actual interaction with the other
parts of x. (A component is a part of an object,
rather than just part of it; see e.g. Tversky 1989, Simons
and Dement 1996.) Clearly, the properties of such restricted relations
may not coincide with those of parthood understood more broadly, and
it will be apparent that pure mereology is only concerned with the
latter.

On the other hand, the English word ‘part’ is
sometimes used in a broader sense, too, for instance to designate the
relation of material constitution, as in (9), or the relation of
mixture composition, as in (10), or the relation of group membership,
as in (11):

(9)

The clay is part of the statue.

(10)

Gin is part of martini.

(11)

The goalie is part of the team.

The mereological status of these relations, however, is
controversial. For instance, although the constitution relation
exemplified in (9) was included by Aristotle in his threefold taxonomy
of parthood (Metaphysics, Δ, 1023b), many contemporary
authors would rather construe it as a sui generis,
non-mereological relation (see e.g. Wiggins 1980, Rea 1995, Baker
1997, Evnine 2011) or else as the relation of identity (Noonan 1993,
Pickel 2010), possibly contingent or occasional identity (Gibbard
1975, Robinson 1982, Gallois 1998). Similarly, the ingredient-mixture
relationship exemplified in (10) is of dubious mereological status, as
the ingredients may undergo significant chemical transformations that
alter the structural characteristics they have in isolation (Sharvy
1983, Bogen 1995, Fine 1995a, Needham 2007). As for cases such as
(11), there is disagreement concerning whether teams and other groups
should be regarded as genuine mereological wholes, and while there are
philosophers who do think so (from Oppenheim and Putnam 1958 to
Quinton 1976, Copp 1984, Martin 1988, and Sheehy 2006), many are
inclined to regard groups as entities of a different sort and to
construe the relation of group membership as distinct from parthood
(see e.g. Simons 1980, Ruben 1983, Gilbert 1989, Meixner 1997,
Uzquiano 2004, Effingham 2010b, and Ritchie 2013 for different
proposals). For all these reasons, here we shall take mereology to be
concerned mainly with the principles governing the relation
exemplified in (1)–(8), leaving it open whether one or more such
broader uses of ‘part’ may themselves be subjected to
mereological treatments of some sort.

Finally, it is worth stressing that mereology assumes no
ontological restriction on the field of ‘part’. In
principle, the relata can be as different as material bodies, events,
geometric entities, or spatio-temporal regions, as in (1)–(8), as well
as abstract entities such as properties, propositions, types, or
kinds, as in the following examples:

(12)

Rationality is part of personhood.

(13)

The antecedent is the ‘if’ part of the conditional.

(14)

The letter ‘m’ is part of the word ‘mereology’.

(15)

Carbon is part of methane.

This is not uncontentious. For instance, to some philosophers the
thought that such abstract entities may be structured mereologically
cannot be reconciled with their being universals. To adapt an example
from Lewis (1986a), if the letter-type ‘m’ is part of the
word-type ‘mereology’, then so is the letter-type
‘e’. But there are two occurrences of ‘e’ in
‘mereology’. Shall we say that the letter is part of the
word twice over? Likewise, if carbon is part of
methane, then so is hydrogen. But each methane
molecule consists of one carbon atom and four hydrogen atoms. Shall we
say that hydrogen is part of methane four times
over? What could that possibly mean? How can one thing be part of
another more than once? These are pressing questions, and the friend
of structured universals may want to respond by conceding that the
relevant building relation is not parthood but, rather, a
non-mereological mode of composition (Armstrong 1986, 1988). However,
other options are open, including some that take the difficulty at
face value from a mereological standpoint (see e.g. Bigelow and
Pargetter 1989, Hawley 2010, Mormann 2010, Bader 2013, and Forrest 2013; see also
D. Smith 2009: §4, K. Bennett 2013, Fisher 2013, and Cotnoir 2013b:
§4, 2015b for explicit discussion of the idea of being part-related “many
times over”). Whether such options are viable may be
controversial. Yet their availability bears witness to the full
generality of the notion of parthood that mereology seeks to
characterize. In this sense, the point to be stressed is
metaphilosophical. For while Leśniewski's and Leonard and
Goodman's original formulations betray a nominalistic stand,
reflecting a conception of mereology as an ontologically parsimonious
alternative to set theory, there is no necessary link between the
analysis of parthood relations and the philosophical position of
nominalism.[5]
As a formal theory (in Husserl's sense of
‘formal’, i.e., as opposed to ‘material’)
mereology is simply an attempt to lay down the general principles
underlying the relationships between an entity and its constituent
parts, whatever the nature of the entity, just as set theory is an
attempt to lay down the principles underlying the relationships
between a set and its members. Unlike set theory, mereology is not
committed to the existence of abstracta: the whole can be as
concrete as the parts. But mereology carries no nominalistic
commitment to concreta either: the parts can be as abstract
as the whole.

Whether this way of conceiving of mereology as a general and
topic-neutral theory holds water is a question that will not be
further addressed here. It will, however, be in the background of much
that follows. Likewise, little will be said about the important
question of whether one should countenance different (primitive)
part-whole relations to hold among different kinds of entity (as urged
e.g. by Sharvy 1980, McDaniel 2004, 2009, and Mellor 2006), or perhaps
even among entities of the same kind (Fine 1994, 2010). Such a
question will nonetheless be relevant to the assessment of certain
mereological principles discussed below, whose generality may be
claimed to hold only in a restricted sense, or on a limited
understanding of ‘part’. For further issues concerning the
alleged universality and topic-neutrality of mereology, see also
Johnston (2005, 2006), Varzi (2010), Donnelly (2011), Hovda (2014),
and Johansson (2015). (Some may even think that there are no parthood
relations whatsoever, e.g., because there are there are no causally
inert non-logical properties or relations, and parthood would be one
such; for a defense of this sort of mereological anti-realism, see
Cowling 2014.)

With these provisos, and barring for the moment the complications
arising from the consideration of intensional factors (such as time
and modalities), we may proceed to review some core mereological
notions and principles. Ideally, we may distinguish here between (a)
those principles that are simply meant to fix the intended meaning of
the relational predicate ‘part’, and (b) a variety of
additional, more substantive principles that go beyond the obvious and
aim at greater sophistication and descriptive power. Exactly where the
boundary between (a) and (b) should be drawn, however, or even whether
a boundary of this sort can be drawn at all, is by itself a matter of
controversy.

The usual starting point is this: regardless of how one feels
about matters of ontology, if ‘part’ stands for the
general relation exemplified by (1)–(8) above, and perhaps also
(12)–(15), then it stands for a partial ordering—a
reflexive, transitive, antisymmetric relation:

(16)

Everything is part of itself.

(17)

Any part of any part of a thing is itself part of that
thing.

(18)

Two distinct things cannot be part of each other.

As it turns out, most theories put forward in the literature
accept (16)–(18). Some misgivings are nonetheless worth mentioning
that may, and occasionally have been, raised against these principles.

Concerning reflexivity (16), two sorts of worry may be
distinguished. The first is that many legitimate senses of
‘part’ just fly in the face of saying that a whole is part
of itself. For instance, Rescher (1955) famously objected to Leonard
and Goodman's theory on these grounds, citing the biologists' use of
‘part’ for the functional subunits of an organism as a
case in point: no organism is a functional subunit of itself. This is
a legitimate worry, but it appears to be of little import. Taking
reflexivity (and antisymmetry) as constitutive of the meaning of
‘part’ simply amounts to regarding identity as a limit
(improper) case of parthood. A stronger relation, whereby nothing
counts as part of itself, can obviously be defined in terms of the
weaker one, hence there is no loss of generality (see Section 2.2
below). Vice versa, one could frame a mereological theory by taking
proper parthood as a primitive instead. As already Lejewski (1957)
noted, this is merely a question of choosing a suitable primitive, so
nothing substantive follows from it. (Of course, if one thinks that
there are or might be objects that are not self-identical,
for instance because of the loss of individuality in the quantum
realm, or for whatever other reasons, then such objects would not be
part of themselves either, yielding genuine counterexamples to
(16). Here, however, we stick to a notion of identity that obeys
traditional wisdom, which is to say a notion whereby identity is an
equivalence relation subject to Leibniz's law.) The second sort of
worry is more serious, for it constitutes a genuine challenge to the
idea that (16) expresses a principle that is somehow constitutive of
the meaning of ‘part’, as opposed to a substantive
metaphysical thesis about parthood. Following Kearns (2011), consider
for instance a scenario in which an enduring wall, W, is shrunk down
to the size of a brick and eventually brought back in time so as to be
used to build (along with other bricks) the original W. Or suppose
wall W is bilocated to my left and my right, and I shrink it to the
size of a brick on the left and then use it to replace a brick from W
on the right. In such cases, one might think that W is part of itself
in a sense in which ordinary walls are not, hence that either parthood
is not reflexive or proper parthood is not irreflexive. For another
example (also by Kearns), if shapes are construed as abstract
universals, then self-similar shapes such as fractals may very well be
said to contain themselves as parts in a sense in which other shapes
do not. Whether such scenarios are indeed possible is by itself a
controversial issue, as it depends on a number of background
metaphysical questions concerning persistence through time, location
in space, and the nature of shapes. But precisely insofar as the
scenarios are not obviously impossible, the generality and
metaphysical neutrality of (16) may be questioned. (Note that those
scenarios also provide reasons to question the generality of many
other claims that underlie the way we ordinarily talk, such as the
claim that nothing can be larger than itself, or
next to itself, or qualitatively different from
itself. Such claims might be even more entrenched in common sense than
the claim that proper parthood is irreflexive, and parthood reflexive;
yet this is hardly a reason to hang on to them at every cost. It
simply shows that our ordinary talk does not take into account
situations that are—admittedly—extraordinary.)

Similar considerations apply to the transitivity principle, (17). On
the one hand, several authors have observed that many legitimate
senses of ‘part’ are non-transitive, fostering the study
of mereologies in which (17) may fail (Pietruszczak 2014). Examples would
include: (i) a biological subunit of a cell is not a part of the
organ(ism) of which that cell is a part; (ii) a handle can be part of
a door and the door of a house, though a handle is never part of a
house; (iii) my fingers are part of me and I am part of the team, yet
my fingers are not part of the team. (See again Rescher 1955 along
with Cruse 1979 and Winston et al. 1987, respectively; for
other examples see Iris et al. 1988, Moltmann 1997, Hossack
2000, Johnston 2002, 2005, Johansson 2004, 2006, and Fiorini
et al. 2014). Arguably,
however, such misgivings stem again from the ambiguity of the English
word ‘part’. What counts as a biological subunit of a cell
may not count as a subunit, i.e., a distinguished part of the
organ, but that is not to say that it is not part of the organ at
all. Similarly, if there is a sense of ‘part’ in which a
handle is not part of the house to which it belongs, or my fingers not
part of my team, it is a restricted sense: the handle is not a
functional part of the house, though it is a functional part
of the door and the door a functional part of the house; my fingers
are not directly part of the team, though they are directly
part of me and I am directly part of the team. (Concerning this last
case, Uzquiano 2004: 136–137, Schmitt 2003: 34, and Effingham
2010b: 255 actually read (iii) as a reductio of the very idea
that the group-membership relation is a genuine case of parthood, as
mentioned above ad (11).) It is obvious that if the
interpretation of ‘part’ is narrowed by additional
conditions, e.g., by requiring that parts make a functional or direct
contribution to the whole, then transitivity may fail. In general, if
x is a φ-part of y and y is a
φ-part of z, x need not be a φ-part of
z: the predicate modifier ‘φ’ may not
distribute over parthood. But that shows the non-transitivity of
‘φ-part’, not of ‘part’, and within a
sufficiently general framework this can easily be expressed with the
help of explicit predicate modifiers (Varzi 2006a; Vieu 2006; Garbacz
2007). On the other hand, there is again a genuine worry that,
regardless of any ambiguity concerning the intended interpretation of
‘part’, (17) expresses a substantive metaphysical thesis
and cannot, therefore, be taken for granted. For example, it turns out
that time-travel and multi-location scenarios such as those mentioned
in relation to (16) may also result in violations of the transitivity
of both parthood (Effingham 2010a) and proper parthood (Gilmore 2009;
Kleinschmidt 2011). And the same could be said of cases that involve
no such exotica. For instance, Gilmore (2014) brings
attention to the popular theory of structured propositions originated
with Russell (1903). Already Frege (1976: 79) pointed out that if the
constituents of a proposition are construed mereologically as (proper)
parts, then we have a problem: assuming that Mount Etna is literally
part of the proposition that Etna is higher than Vesuvius, each
individual piece of solidified lava that is part of Etna would also be
part of that proposition, which is absurd. The worse for Russell's
theory of structured propositions, said Frege. The worse, one could
reply, for the transitivity of parthood (short of claiming that the
argument involves yet another equivocation on ‘part
of’).

Concerning the antisymmetry postulate (18), the picture is even more
complex. For one thing, some authors maintain that the relationship
between an object and the stuff it is made of provides a perfectly
ordinary counterexample of the antisymmetry of parthood: according to
Thomson (1998), for example, a statue and the clay that constitutes it
are part of each other, yet distinct. This is not a popular view: as
already mentioned, most contemporary authors would either deny that
material constitution is a relation of parthood or else treat it as
improper parthood, i.e., identity, which is trivially
antisymmetric (and symmetric). Moreover, those who regard constitution
as a genuine case of proper parthood tend to follow Aristotle's
hylomorphic conception and deny that the relation also holds in the
opposite direction: the clay is part of the statue but not vice versa
(see e.g. Haslanger 1994, Koslicki 2008). Still, insofar as Thomson's
view is a legitimate option, it represents a challenge to the putative
generality of (18). Second, one may wonder about the possibility of
unordinary cases of symmetric parthood relationships. Sanford
(1993: 222) refers to Borges's Aleph as a case in point: “I saw
the earth in the Aleph and in the earth the Aleph once more and the
earth in the Aleph …”. In this case, a plausible reply is
simply that fiction delivers no guidance to conceptual investigations:
conceivability may well be a guide to possibility, but literary
fantasy is by itself no evidence of conceivability (van Inwagen 1993:
229). Perhaps the same could be said of Fazang's Jeweled Net of Indra,
in which each jewel has every other jewel as part (Jones
2012). However, other cases seem harder to dismiss. Surely the
Scholastics were not merely engaging in literary fiction when arguing
that each person of the Trinity is a proper part of God, and yet also
identical with God (see e.g. Abelard, Theologia christiana,
bk. III). And arguably time travel is at least conceivable, in which
case again (18) could fail: if time-traveling wall W ends up being one
of the bricks that compose (say) its own bottom half, H, then we have
a conceivable scenario in which W is part of H and H is part of W
while W ≠ H (Kleinschmidt 2011). Third, it may be argued that
antisymmetry is also at odds with theories that have been found
acceptable on quite independent grounds. Consider again the theory of
structured propositions. If A is the proposition that the
universe exists—where the universe is something of which
everything is part—and if A is true, then on such a
theory the universe would be a proper part of A; and since
A would in turn be a proper part of U, antisymmetry would be
forfeit (Tillman and Fowler 2012). Likewise, if A is the
proposition that B is true, and B the proposition
that A is contingent, then again A and B
would be part of each other even though A ≠ B
(Cotnoir 2013b). Finally, and more generally, it may be observed that
the possibility of mereological loops is to be taken seriously for the
same sort of reasons that led to the development of non-well-founded
set theory, i.e., set theory tolerating cases of self-membership and,
more generally, of membership circularities (Aczel 1988; Barwise and
Moss 1996). This is especially significant in view of the possibility
of reformulating set theory itself in mereological terms—a
possibility that is extensively worked out in the works of Bunt (1985)
and especially Lewis (1991, 1993b). For all these reasons, the
antisymmetry postulate (18) can hardly be regarded as constitutive of
the basic meaning of ‘part’, and some authors have begun
to engage in the systematic study of “non-well-founded
mereologies” in which (18) may fail (Cotnoir 2010; Cotnoir and
Bacon 2012; Obojska 2013).

In the following we aim at a critical survey of mereology as standardly
understood, so we shall mainly confine ourselves to theories that do
in fact accept the antisymmetry postulate along with both reflexivity
and transitivity. However, the above considerations should not be
dismissed. On the contrary, they are crucially relevant in assessing
the scope of mereology and the degree to which its standard
formulations and extensions betray intuitions that may be found too
narrow, false, or otherwise problematic. Indeed, they are crucially
relevant also in assessing the ideal desideratum mentioned at the
beginning of this section—the desideratum of a neat demarcation
between core principles that are simply meant to fix the intended meaning
of ‘part’ and principles that reflect more substantive
theses concerning the parthood relation. Classical mereology takes the
former to include the threefold claim that ‘part’ stands
for a reflexive, transitive and antisymmetric relation, but this is
not to say that “anyone who seriously disagrees with them had
failed to understand the word” (Simons 1987: 11), just as
departure from the basic principles of classical logic need not amount
to a “change of subject” (Quine 1970: 81). And just as the
existence of widespread and diversified disagreement concerning the
laws of logic may lead one to conclude that “for all we know,
the only inference left in the intersection of (unrestricted)
all logics might be the identity inference: From
A to infer A” (Beall and Restall 2006: 92), so
one might take the above considerations and the corresponding
development of non-classical mereologies to indicate that there may be
“no reason to assume that any useful core mereology […]
functions as a common basis for all plausible metaphysical
theories” (Donnelly 2011: 246).

It is convenient at this point to introduce some degree of
formalization. This avoids ambiguities stemming from ordinary language
and facilitates comparisons and developments. For definiteness, we
assume here a standard first-order language with identity, supplied
with a distinguished binary predicate constant, ‘P’, to be
interpreted as the parthood
relation.[6]
Taking the underlying logic to
be the classical predicate calculus with
identity,[7]
the requisites on parthood discussed in Section 2.1 may then
be regarded as forming a first-order theory characterized by
the following proper axioms for ‘P’:

(P.1)

Reflexivity
Pxx

(P.2)

Transitivity
(Pxy ∧ Pyz) → Pxz

(P.3)

Antisymmetry
(Pxy ∧ Pyx) → x=y.

(Here and in the following we simplify notation by dropping all
initial universal quantifiers. Unless otherwise specified, all
formulas are to be understood as universally closed.) We may call such
a theory Core Mereology—M for
short[8]—since
it represents the common starting point of all standard theories.

Given (P.1)–(P.3), a number of additional mereological predicates
can be introduced by definition. For example:

(19)

Equality
EQxy =df Pxy ∧ Pyx

(20)

Proper Parthood
PPxy =df Pxy ∧ ¬x=y

(21)

Proper Extension
PExy =df Pyx ∧ ¬x=y

(22)

Overlap
Oxy =df ∃z(Pzx ∧ Pzy)

(23)

Underlap
Uxy =df ∃z(Pxz ∧ Pyz).

An intuitive model for these relations, with ‘P’
interpreted as spatial inclusion, is given in Figure 1.

Note that ‘Uxy’ is bound to hold if one assumes
the existence of a “universal entity” of which everything
is part. Conversely, ‘Oxy’ would always hold if
one assumed the existence of a “null item” that is part of
everything. Both assumptions, however, are controversial and we shall
come back to them below.

Note also that the definitions imply (by pure logic) that EQ, O, and U
are all reflexive and symmetric; in addition, EQ is also
transitive—an equivalence relation. By contrast, PP and PE are
irreflexive and asymmetric, and it follows from (P.2) that both are
also transitive—so they are strict partial orderings. Since the
following biconditional is also a straightforward consequence of the
axioms (specifically, of P.1),

(24)

Pxy ↔ (PPxy ∨ x=y),

it should now be obvious that one could in fact use proper parthood as
an alternative starting point for the development of classical mereology, using
the right-hand side of (24) as a definiens for ‘P’. This
is, for instance, the option followed in Simons (1987), as also in
Leśniewski’s original theory (1916), where the
partial ordering axioms for ‘P’ are replaced by the strict
ordering axioms for
‘PP’.[9]
Ditto for ‘PE’,
which was in fact the primitive relation in Whitehead's (1919)
semi-formal treatment of the mereology of events (and which is just
the converse of ‘PP’). Other options are in principle
possible, too. For example, Goodman (1951) used ‘O’ as a
primitive and Leonard and Goodman (1940) used its
opposite:[10]

(25)

Disjointness
Dxy =df ¬Oxy.

However, the relations corresponding to such predicates are strictly
weaker than PP and PE and no biconditional is provable
in M that would yield a corresponding definiens of
‘P’ (though one could define ‘P’ in terms of
‘O’ or ‘D’ in the presence of further axioms;
see below ad (61)). Thus, other things being equal,
‘P’, ‘PP’, and ‘PE’ appear to be
the only reasonable options. Here we shall stick to
‘P’, referring to J. Parsons (2014) for further discussion.

Finally, note that identity could itself be introduced by definition,
due to the following obvious consequence of the antisymmetry postulate
(P.3):

(26)

x=y ↔ EQxy.

Accordingly, theory M could be formulated in a pure
first-order language by assuming (P.1) and (P.2) and replacing (P.3)
with the following variant of the Leibniz axiom schema for identity (where
φ is any formula in the language):

(P.3′)

Indiscernibility
EQxy → (φx ↔ φy).

One may in fact argue on these grounds that the parthood relation is
in some sense conceptually prior to the identity relation (as in
Sharvy 1983: 234), and since ‘EQ’ is not definable in
terms of ‘PP’ or ‘PE’ alone except in the
presence of stronger axioms (see below ad (27)), the argument
would also provide evidence in favor of ‘P’ as the most
fundamental primitive. As we shall see in Section 3.2, however, the
link between parthood and identity is philosophically problematic. In
order not to compromise our exposition, we shall therefore keep to a
language containing both ‘P’ and ‘=’ as
primitives. This will also be convenient in view of the previous
remarks concerning the controversial status of Antisymmetry, on which
(26) depends.

The last remark is also relevant to the definition of ‘PP’
given above. That is the classical definition used by Leśniewski
and by Leonard and Goodman and corresponds verbatim to the intuitive
characterization of proper parthood used in the previous
section. However, in some treatments (including earlier versions of
this entry[11]),
‘PP’ is defined directly in terms of ‘P’,
without using identity, as per the following variant of (20):

(20′)

(Strict) Proper Parthood
PPxy =df Pxy ∧ ¬Pyx.

(See e.g. Goodman 1951: 35; Eberle 1967: 272; Simons 1991a: 286; Casati
and Varzi 1999: 36; Niebergall 2011: 274). Similarly for
‘PE’. In M the difference is immaterial,
since the relevant definientia are provably equivalent. But the
equivalence in question depends crucially on Antisymmetry. Absent
(P.3), the second definition is strictly stronger: any two things that
are mutually P-related would count as proper parts of each other
according to (20) but not, obviously, according to (20′), which
forces PP to be asymmetric. Indeed, in the presence of (P.1) and (P.2)
the latter definition is still strong enough to deliver a strict
partial ordering, whereas (20) does not even yield a transitive relation
unless (P.3) is assumed.[12] Another
important difference is that, absent (P.3), the biconditional in (24)
continues to hold only if ‘PP’ is defined as in (20); if
(20′) is used instead, the left-to-right direction fails
whenever x and y are distinct mutual parts. In view
of the above remarks concerning the doubtful status of (P.3), it is
therefore convenient to work with the weaker definition. Standardly it
makes no difference, but some of the definitions and results presented
below would not extend to non-well-founded mereology if (20′)
were used instead. (See e.g. Cotnoir 2010 and Gilmore 2015.)
Furthermore, since both definitions force PP to be irreflexive, it
should be noted that the only way to develop a non-well-founded
mereology that allows for strict mereological loops, i.e., things that
are proper part of themselves, is to rely on yet another definition or
else take ‘PP’ as a primitive (as in Cotnoir and Bacon
2012, where PP is axiomatized as transitive but neither irreflexive
nor asymmetric).

M is standardly viewed as embodying the common core
of any mereological theory. Not just any partial ordering qualifies as
a part-whole relation, though, and establishing what further
principles should be added to (P.1)–(P.3) is precisely the
question a good mereological theory is meant to answer. It is here
that philosophical issues begin to multiply, over and above the
general concerns mentioned in Section 2.1.

Generally speaking, such further principles may be divided into two
main groups. On the one hand, one may extend M by
means of decomposition principles that take us from a whole
to its parts. For example, one may consider the idea that whenever
something has a proper part, it has more than one—i.e., that
there is always some mereological difference (a “remainder”)
between a whole and its proper parts. This need not be true in every
model for M: a world with only two items, only one of
which is part of the other, would be a counterexample, though not one
that could be illustrated with the sort of geometric diagram used in
Figure 1. On the other hand, one may extend M by
means of composition principles that go in the opposite
direction—from the parts to the whole. For example, one may
consider the idea that whenever there are some things, there exists a
whole that consists exactly of those things—i.e., that there is
always a mereological sum (or “fusion”) of two or
more parts. Again, this need not be true in a model
for M, and it is a matter of much controversy whether the
idea should hold unrestrictedly.

Let us begin with the first sort of extension. And let us start by
taking a closer look at the intuition according to which a whole
cannot be decomposed into a single proper part. There are various ways
in which one can try to capture this intuition. Consider the following
(from Simons 1987: 26–28):

The first principle, (P.4a), is a literal rendering of the
idea in question: every proper part must be accompanied by
another. However, there is an obvious sense in which (P.4a)
only captures the letter of the idea, not the spirit: it rules out the
unintended model mentioned above (see Figure 2, left) but not, for
example, an implausible model with an infinitely descending chain in
which the additional proper parts do not leave any remainder at all (Figure
2, center).

The second principle, (P.4b), is stronger: it rules out
both models as unacceptable. However, (P.4b) is still too
weak to capture the intended idea. For example, it is satisfied by a
model in which a whole can be decomposed into several proper parts all
of which overlap one another (Figure 2, right), and it may be argued
that such models do not do justice to the meaning of ‘proper
part’: after all, the idea is that the removal of a proper part
should leave a remainder, but it is by no means clear what would be
left of x once z (along with its parts) is
removed.

It is only the third principle, (P.4), that appears to provide a full
formulation of the idea that a whole cannot be decomposed into a single proper
part. According to this principle, every proper part must be
“supplemented” by another, disjoint part, and it
is this last qualification that captures the notion of a
remainder. Should (P.4), then, be incorporated into M
as a further fundamental principle on the meaning of
‘part’?

Most authors (beginning with Simons himself) would
say so. Yet here there is room for genuine disagreement. In fact, it
is not difficult to conceive of mereological scenarios that violate
not only (P.4), but also (P.4b) and even (P.4a).
A case in point would be Brentano's (1933) theory of accidents,
according to which a mind is a proper part of a thinking mind even
though there is nothing to make up for the difference. (See Chisholm
1978, Baumgartner and Simons 1993.) Similarly, in Fine's (1982) theory
of qua-objects, every basic object (John) qualifies as the
only proper part of its incarnations (John qua philosopher,
John qua husband, etc.). Another interesting example is
provided by Whitehead's (1929) theory of extensive connection, where
no boundary elements are included in the domain of quantification: on
this theory, a topologically closed region includes its open interior
as a proper part in spite of there being no boundary elements to
distinguish them—the domain only consists of extended
regions. (See Clarke 1981 for a rigorous formulation, Randell et
al. 1992 for developments.) Finally, consider the view, arguably
held by Aquinas, according to which the human person survives physical
death along with her soul (see Brown 2005 and Stump 2006,
pace Toner 2009). On the understanding that persons are
hylomorphic composites, and that two things cannot become one, the
view implies that upon losing her body a person will continue to
exist, pre-resurrection, with only one proper part—the
soul. (This is also the view of some contemporary philosophers; see
e.g. Oderberg 2005 and Hershenov and Koch-Hershenov 2006.) Indeed, any
case of material coincidence resulting from mereological diminution,
as in the Stoic puzzle of Deon and Theon (Sedley 1982) and its modern
variant of Tibbles and Tib (Wiggins 1968), would seem to be at odds
with Supplementation: after the diminution, there is nothing that
makes up for the difference between what was a proper part and the
whole with which it comes to coincide, short of holding that the part
has become identical to the whole (Gallois 1998), or has ceased to
exist (Burke 1994), or did not exist in the first place (van Inwagen
1981). One may rely on the intuitive appeal of (P.4) to discard all of
the above theories and scenarios as implausible. But one may as well
turn things around and regard the plausibility of such theories as a
good reason not to accept (P.4) unrestrictedly, as argued e.g. by
D. Smith (2009), Oderberg (2012), and Lowe (2013). As things stand, it
therefore seems appropriate to regard such a principle as providing a
minimal but substantive addition to (P.1)–(P.3), one that goes
beyond the basic characterization of ‘part’ provided by
M. We shall label the resulting mereological theory
MM, for Minimal Mereology.

Actually MM is now redundant, as Supplementation
turns out to entail Antisymmetry so long as parthood is transitive and
reflexive: if x and y were proper parts of each
other, contrary to (P.3), then every z that is part of one
would also be part of—hence overlap—the other, contrary to
(P.4). For ease of reference, we shall continue to treat (P.3) as an
axiom. But the entailment is worth emphasizing, for it explains why
Supplementation tends to be explicitly rejected by those who
do not endorse Antisymmetry, over and above the more classical
examples mentioned above. For instance, whoever thinks that a statue
and the corresponding lump of clay are part of each other will find
Supplementation unreasonable: after all, such parts are coextensive;
why should we expect anything to be left over when, say, the clay is
“subtracted” from the statue? (Donnelly 2011:
230). Indeed, Supplementation has recently run into trouble also
independently of its link with Antisymmetry, especially in the context
of time-travel and multilocation scenarios such as those already
mentioned in connection with each of (P.1)–(P.3) (see Effingham
and Robson 2007, Gilmore 2007, Eagle 2010, Kleinschmidt 2011, Daniels 2014). As a
result, a question that is gaining increasing attention is whether
there are any ways of capturing the supplementation intuition that are
strong enough to rule out the models of Figure 2 and yet sufficiently
weaker than (P.4) to be acceptable to those who do not endorse some
M-axiom or other—be it Antisymmetry,
Transitivity, or Reflexivity.

Two sorts of answer may be offered in this regard (see
e.g. Gilmore 2015). The first is to weaken the Supplementation
conditional by strengthening the antecedent. For instance, one may
simply rephrase (P.4) in terms of the stricter notion of proper
parthood defined in (20′), i.e., effectively:

(P.4c)

Strict Supplementation (Pxy ∧
¬Pyx) → ∃z(Pzy ∧
¬Ozx).

In M this is equivalent to (P.4). Yet it is logically
weaker, and it is easy to see that this suffices to block the
entailment of (P.3) even in the presence of (P.1)–(P.2) (just
consider a two-element model with mutual parthood, as in Figure 3,
left). Still, (P.4c) is sufficiently stronger than (P.4a) and
(P.4b) to rule out all three patterns in Figure 2, and it
obviously preserves the spirit of (P.4)—if not the letter. The
second sort of answer is to weaken Supplementation by adjusting the
consequent. There are various ways of doing this, the most natural of
which appears to be the following:

(P.4d)

Quasi-supplementation PPxy →
∃z∃w(Pzy ∧ Pwy
∧ ¬Ozw).

Again, this principle is stronger than (P.4a) and
(P.4b), since it rules out all patterns in Figure 2, and in
M it is equivalent to (P.4). Indeed,
(P.4d) says, literally, that if something has a proper
part, then it has at least two disjoint parts, which Simons
(1987: 27) takes to express the same intuition captured by (P.4). Yet
(P.4d) is logically weaker than (P.4), since it admits the
non-antisymmetric model in Figure 3, middle, and for that reason it
may be deemed more suitable in the context of theories that violate
(P.3). Note also that (P.4d) does not admit the symmetric model
on Figure 3, left, so in a way it is stronger than
(P.4c). In another way, however, it is weaker, since it
admits the model in Figure 3, right, which (P.4c) rules out
(and which someone who thinks that, say, the clay is part of the
statue, but not vice versa, might want to retain).

Figure 3. More unsupplemented patterns.

There are other options, too. For instance, in some standard
treatments, the Supplementation principle (P.4) is formulated using
‘PP’ also in the consequent:

(P.4′)

Proper Supplementation
PPxy → ∃z(PPzy ∧ ¬Ozx).

In M this is once again equivalent to (P.4), but the equivalence
depends on Reflexivity and Symmetry. Absent (P.1) or (P.2),
(P.4′) is logically stronger. Yet again one may rely on the alternative
definition of ‘PP’ to obtain variants of (P.4′) that
are stronger than (P.4c) and weaker than (P.4). Similarly
for (P.4d), which may be further weakened or strengthened
by tampering with the parthood predicates occurring in the antecedent
and in the consequent.

We may also ask the opposite question: Are there any stronger ways of
expressing the supplementation intuition besides (P.4)? In classical
mereology, the standard answer is in the affirmative, the main
candidate being the following:

(P.5)

Strong Supplementation
¬Pyx → ∃z(Pzy ∧ ¬Ozx).

Intuitively, this says that if an object fails to include
another among its parts, then there must be a remainder, something
that makes up for the difference. It is easily
seen that, given M, (P.5) implies (P.4), so any
M-theory violating (P.4) will a fortiori violate (P.5). For
instance, on Whitehead's boundary-free theory of extensive connection,
a closed region is not part of its interior even though each part of
the former overlaps the latter. More generally, the entailment holds
as long as parthood is antisymmetric (see again Figure 3, center, for a
non-antisymmetric counterexample). However, the converse is not
true. The diagram in Figure 4 illustrates an M-model
in which (P.4) is satisfied, since each proper part counts as a
supplement of the other; yet (P.5) is false.

Figure 4. A supplemented model violating Strong Supplementation.

The theory obtained by adding (P.5) to (P.1)–(P.3) is thus a proper
extension of MM. We label this stronger
theory EM, for Extensional Mereology, the
attribute ‘extensional’ being justified precisely by the
exclusion of countermodels that, like the one in Figure 4, contain
distinct objects with the same proper parts. In fact, it is a theorem
of EM that no composite objects with the
same proper parts can be distinguished:

(27)

(∃zPPzx ∨ ∃zPPzy)
→ (x=y ↔ ∀z(PPzx
↔ PPzy)).

(The analogue for ‘P’ is already provable in
M, since P is reflexive and antisymmetric.) This goes
far beyond the intuition that lies behind the basic Supplementation
principle (P.4). Does it go too far?

On the face of it, it is not difficult to envisage scenarios that
would correspond to the diagram in Figure 4. For example, we may
take x and y to be the sets
{{z},
{z, w}} and
{{w},{z, w}},
respectively (i.e., the ordered pairs
⟨z, w⟩ and
⟨w, z⟩), interpreting
‘P’ as the ancestral of the improper membership relation
(i.e., of the union of ∈ and =). But sets are abstract entities,
and the ancestral relation does not generally satisfy (P.4) (the
singleton of the empty set, for instance, or the singleton of
any urelement, would have only one proper part on the
suggested construal of ‘P’). Can we also envisage similar
scenarios in the domain of concrete, spatially extended entities,
granting (P.4) in its generality? Admittedly, it is difficult
to picture two concrete objects mereologically structured as
in Figure 4. It is difficult, for example, to draw two
extended objects composed of the same proper parts because drawing
something is drawing its proper parts; once the parts are
drawn, there is nothing left to be done to get a drawing of the
whole. Yet this only proves that pictures are biased towards
(P.5). Are there any philosophical reasons to resist the extensional
force of (P.5) beyond the domain of abstract entities, and in the
presence of (P.4)?

Two sorts of reason are worth examining. On the one hand, it is
sometimes argued that sameness of proper parts is not
sufficient for identity. For example, it is argued
that: (i) two words can be made up of the same letters (Hempel 1953:
110; Rescher 1955: 10), two tunes of the same notes (Rosen and Dorr
2002: 154), and so on; or (ii) the same flowers can compose a nice
bunch or a scattered bundle, depending on the arrangements of the
individual flowers (Eberle 1970: §2.10); or (iii) two groups can
have co-extensive memberships, say, the Library Committee and the
Philosophy Department football team (Simons 1987: 114; Gilbert 1989:
273); or (iv) a cat must be distinguished from the corresponding
amount of feline tissue, for the former can survive the annihilation
of certain parts (the tail, for instance) whereas the latter cannot by
definition (Wiggins 1968; see also Doepke 1982, Lowe 1989, Johnston
1992, Baker 1997, Meirav 2003, Sanford 2003, and Crane 2012, inter
alia, for similar or related arguments). On the other hand, it is
sometimes argued that sameness of parts is not necessary for
identity, as some entities may survive mereological change. If a cat
survives the annihilation of its tail, then the tailed cat (before the
accident) and the tailless cat (after the accident) are numerically
the same in spite of their having different proper parts (Wiggins
1980). If any of these arguments is accepted, then clearly (27) is too
strong a principle to be imposed on the parthood relation. And since
(27) follows from (P.5), it might be concluded that
EM is on the wrong track.

Let us look at these objections separately. Concerning the necessity
aspect of mereological extensionality, i.e., the left-to-right
conditional in the consequent of (27),

(28)

x=y → ∀z(PPzx ↔ PPzy),

it is perhaps enough to remark that the difficulty is not peculiar
to extensional mereology. The objection proceeds from the
consideration that ordinary entities such as cats and other living
organisms (and possibly other entities as well, such as statues and
ships) survive all sorts of gradual mereological change. This a
legitimate thought, lest one be forced into some form of
“mereological essentialism” (Chisholm 1973, 1975, 1976;
Plantinga 1975; Wiggins 1979). However, the same can be said of other
types of change as well: bananas ripen, houses deteriorate, people
sleep at night and eat at lunch. How can we say that they are the same
things, if they are not quite the same? Indeed, (28) is essentially an
instance of the identity axiom schema

(ID)

x=y → (φx ↔ φy),

and it is well known that this axiom schema runs into trouble when
‘=’ is given a diachronic reading. (See the
entries on
change
and
identity over time.)
The problem is a general one. Whatever the solution, it will therefore
apply to the case at issue as well, and in this sense the
above-mentioned objection to (28) can be disregarded. For example, the
problem would dissolve immediately if the variables in (28) were taken
to range over four-dimensional entities whose parts may extend in time
as well as in space (Heller 1984, Lewis 1986b, Sider 2001), or if
identity itself were construed as a contingent relation that may hold
at some times or worlds but not at others (Gibbard 1975, Myro 1985,
Gallois 1998). Alternatively, on a more traditional, three-dimensional
conception of material objects, the problem of change is often
accounted for by relativizing properties and relations to times,
rewriting (ID) as

(ID′)

x=y →
∀t(φtx ↔
φty).

(This may be understood in various ways; see e.g. the papers in
Haslanger and Kurtz 2006, Part III.) If so, then again the specific worry
about (28) would dissolve, as the relativized version of (P.5) would
only warrant the following variant of the conditional in question:

(28′)

x=y →
∀t∀z(PPtzx ↔
PPtzy).

(See Thomson 1983, Simons 1987: §5.2, Masolo 2009, Giaretta
and Spolaore 2011; see also Kazmi 1990 and Hovda 2013 for tensed
versions of this strategy.) The need to relativize parthood to time,
and perhaps to other parameters such as space, possible worlds, etc.,
has recently been motivated also on independent grounds, from the
so-called “problem of the many” (Hudson 2001) to material
constitution (Bittner and Donnelly 2007), modal realism (McDaniel
2004), vagueness (Donnelly 2009), relativistic spacetime (Balashov
2008), or the general theory of location (Gilmore 2009, Donnelly
2010). One way or the other, then, such revisions may be regarded as
an indicator of the limited ontological neutrality of extensional
mereology. But their independent motivation also bears witness to the
fact that controversies about (28) stem from genuine and fundamental
philosophical conundrums and cannot be assessed by appealing to our
intuitions about the meaning of ‘part’.

The worry about the sufficiency aspect of mereological extensionality, i.e.,
the right-to-left conditional in the consequent of (27),

(29)

∀z(PPzx ↔ PPzy) → x=y,

is more to the point. However, here too there are various ways of
responding on behalf of EM. Consider counterexample
(i)—say, two words made up of the same letters, as in
‘else’ and ‘seel’. If these are taken as
word-types, a lot depends on how exactly one construes such things
mereologically, and one might simply dismiss the challenge by
rejecting, or improving on, the dime-store thought that word-types are
letter-type composites (see above ad (14)). Indeed, if they
were, then word-types would not only violate extensionality, hence the
Strong Supplementation principle (P.5); they would violate the basic
Supplementation principle (P.4), since ‘seel’ (for
instance) would contain a proper part (the string ‘ee’)
that consists of a single proper part (the letter ‘e’). On
the other hand, if the items in question are taken as word-tokens,
then presumably they are made up of distinct letter-tokens, so again
there is no violation of (29), hence no reason to reject (P.5) on
these grounds. Of course, we may suppose that one of the two
word-tokens is obtained from the other by rearranging the same
letter-tokens. If so, however, the issue becomes once again one of
diachronic non-identity, with all that it entails, and it is not
obvious that we have a counterexample to (29). (See Lewis 1991: 78f.)
What if our letter-tokens are suitably arranged so as to form both
words at the same time? For example, suppose they are arranged in a
circle (Simons 1987: 114). In this case one might be inclined to say
that we have a genuine counterexample. But one may equally well insist
that we have got just one circular inscription that, curiously, can be
read as two different words depending on where we start. Compare: I
draw a rabbit that to you looks like a duck. Have I thereby made two
drawings? I write ‘p’ on my office glass door; from the
outside you read ‘q’. Have I therefore produced two
letter-tokens? And what if Mary joins you and reads it upside down;
have I also written the letter ‘b’? Surely then I have
also written the letter ‘d’, as my upside-down office mate
John points out. This multiplication of entities seems
preposterous. There is just one thing there, one inscription, and what
it looks (or mean) to you or me or Mary or John is irrelevant to what
that thing is. Similarly—it may be argued—there is just
one inscription in our example, a circular display of four
letter-tokens, and whether we read it as an
‘else’-inscription or a ‘seel’-inscription is
irrelevant to its mereological structure. (Varzi 2008)

Case (ii)—the flowers—is not significantly different. The
same, concrete flowers cannot compose a nice bunch and a scattered
bundle at the same time. Similarly for many other cases of
this sort that may come to mind, including much less frivolous
prima facie counterexamples offered by the natural
sciences—from the different phases of matter (solids, liquids,
and gases) to the different possibilities of chemical binding; see
e.g. Harré and Llored (2011, 2013) and Sukumar (2013). (Not all
cases are so easily dismissed, though. In particular, several
authors—from Maudlin 1998 to Krause 2011—have argued that
the world of quantum mechanics provides genuine type-(ii)
counterexamples to extensionality. A full treatment of such arguments
goes beyond the scope of this entry, but see e.g. Calosi et al.
2011 and Calosi and Tarozzi 2014 for counter-arguments.)

Case (iii) is more delicate, as it depends on one's metaphysics of
such things as committees, teams, and groups generally. If one denies
that the relevant structural relation is a genuine case of parthood
(see Section 1, ad (11)), then of course the counterexample
misfires. If, on the other hand, one takes groups to be bona
fide mereological composites—and composites consisting of
enduring persons as opposed to, say, person-stages, as in Copp
(1984)—then a lot depends on one's reasons to treat groups with
co-extensive memberships as in fact distinct. Typically such reasons
are just taken for granted, as if the distinctness were obvious. But
sometimes informal arguments are offered to the effect that, say, the
coextensive Library Committee and football team must be distinguished
insofar as they have different persistence conditions, or different
properties broadly understood. For instance, the players of the team
can change even though the Committee remains the same, or one group
can be dismantled even though the other continues to operate, or one
group has different legal obligations than the other, and so on (see
e.g. Moltmann 1997). If so, then case (iii) becomes relevantly similar
to case (iv). There, too, the intuition is that a living animal such
as a cat is something “over and above” the mere lump of
feline tissue that constitutes its body—that they have different
survival conditions and, hence, different properties—so it
appears that here we have a genuine counterexample to mereological
extensionality (via Leibniz's Law). It is for similar reasons that
some philosophers are inclined to treat a vase and the corresponding
lump of clay as distinct in spite of their sharing the same proper
parts—possibly even the same improper parts, contrary to (P.3),
as seen in Section
2.2.[14]
Two responses may nonetheless be offered in
such cases on behalf of EM (besides rejecting the
intuition in question on the basis of a specific metaphysics of
persistence).

Focusing on (iv), the first response is to insist that, on the face of
it, a cat and the corresponding lump of feline tissue (or a statue and
the lump of clay that constitutes it) do not share the same proper
parts after all. For, on the one hand, if one believes that at least
one such thing, x, is part of the other, y, then it
must be a proper part; and insofar as nothing can be a proper part of
itself, it follows immediately that such things do not in fact
constitute a counterexample to (29). (This would also follow from
Supplementation, as emphasized e.g. in Olson 2006, since the
assumption that x and y have the same proper parts
entails that no part of y is disjoint from x, at
least so long as parthood is reflexive; but there is no need to
invoke (P.4) here.) On the other hand, if one believes that neither
x nor y is part of the other, then presumably the
same belief will also apply to some of their proper parts—say,
the cat's tail and the corresponding lump of tissue. And if the tail
is not part of that lump, then presumably it is also not part
of the larger lump of tissue that constitutes the whole cat (as
explicitly acknowledged by some anti-extensionalists, e.g. Lowe 2001:
148 and Fine 2003: 198, n. 5, though see Hershenov 2008 for
misgivings). Thus, again, it would appear that x and
y do not have the same proper parts after all and do not,
therefore, constitute a counterexample to (29). (For more on this line
of argument, see Varzi 2008.)

The second and more general response on behalf of EM
is that the appeal to Leibniz's law in this context is
illegitimate. Let ‘Tibbles’ name our cat and
‘Tail’ its tail, and grant the truth of

(30)

Tibbles can survive the annihilation of Tail.

There is, indeed, an intuitive sense in which the following is also
true:

(31)

The lump of feline tissue constituting Tail and the rest of
Tibbles's body cannot survive the annihilation of Tail.

However, this intuitive sense corresponds to a de dicto
reading of the modality, where the definite description in (31) has narrow
scope:

(31a)

In every possible world, the lump of feline tissue constituting
Tail and the rest of Tibbles's body ceases to exist if Tail is
annihilated.

On this reading, (31) is hardly negotiable. Yet this is irrelevant in
the present context, for (31a) does not amount to an ascription of a
modal property and cannot be used in connection with Leibniz's
law. (Compare: 8 is necessarily even; the number of planets might have
been odd; hence the number of planets is not 8.) On the other hand,
consider a de re reading of (31), where the definite
description has wide scope:

(31b)

The lump of feline tissue constituting Tail and the rest of
Tibbles's body is such that, in every possible world, it ceases to
exist if Tail is annihilated.

On this reading, the appeal to Leibniz's law would be legitimate
(modulo any concerns about the status of modal properties) and one
could rely on the truth of (30) and (31) (i.e., (31b)) to conclude
that Tibbles is distinct from the relevant lump of feline
tissue. However, there is no obvious reason why (31) should be
regarded as true on this reading. That is, there is no obvious reason
to suppose that the lump of feline tissue that in the actual world
constitutes Tail and the rest of Tibbles's body—that
lump of feline tissue that is now resting on the carpet—cannot
survive the annihilation of Tail. Indeed, it would appear that any
reason in favor of this claim vis-à-vis the truth of
(30) would have to presuppose the distinctness of the
entities in question, so no appeal to Leibniz's law would be
legitimate to determine the distinctess (on pain of
circularity). This is not to say that the putative counterexample to
(29) is wrong-headed. But it requires genuine metaphysical work to
establish it and it makes the rejection of extensionality, and with it
the rejection of the Strong Supplementation principle (P.5), a matter of genuine
philosophical controversy. (Similar remarks would apply to any
argument intended to reject extensionality on the basis of competing
modal intuitions regarding the possibility of mereological
rearrangement, rather than mereological change, as
with the flowers example. On a de re reading, the claim that
a bunch of flowers could not survive rearrangement of the
parts—while the aggregate of the individual flowers composing it
could—must be backed up by a genuine metaphysical theory about
those entities. For more on this general line of defense on behalf of
(29), see e.g. Lewis 1971: 204ff, Jubien 1993: 118ff, and Varzi 2000:
291ff. See also King's 2006 reply to Fine 2003 for a more general
diagnosis of the semantic mechanisms at issue here.)

This says that if y is not part of x, there exists
something that comprises exactly those parts of y that are
disjoint from x—something we may call
the difference or relative complement
between y and x. It is easily checked that this
principle implies (P.5). On the other hand, the diagram in Figure 5
shows that the converse does not hold: there are two parts
of y in this diagram that do not overlap x,
namely z and w, but there is
nothing that consists exactly of such parts, so we have a model of
(P.5) in which (P.6) fails.

Figure 5. A strongly supplemented model violating Complementation.

Any misgivings about (P.5) may of course be raised against (P.6). But
what if we agree with the above arguments in support of (P.5)? Do they
also give us reasons to accept the stronger principle (P.6)? The
answer is in the negative. Plausible as it may initially sound, (P.6)
has consequences that even an extensionalist may not be willing to
accept. For example, it may be argued that although the base and the
stem of this wine glass jointly compose a larger part of the glass
itself, and similarly for the stem and the bowl, there is nothing
composed just of the base and the bowl (= the difference between the
glass and the stem), since these two pieces are standing apart. More
generally, it appears that (P.6) would force one to accept the
existence of a wealth of “scattered” entities, such as the
aggregate consisting of your nose and your thumbs, or the aggregate of
all mountains higher than Mont Blanc. And since V. Lowe (1953), many
authors have expressed discomfort with such
entities regardless of extensionality. (One philosopher who
explicitly accepts extensionality but feels uneasy about scattered
entities is Chisholm 1987.) As it turns out, the extra strength of
(P.6) is therefore best appreciated in terms of the sort of
mereological aggregates that this principle would force us to accept,
aggregates that are composed of two or more parts of a given
whole. This suggests that any additional misgivings about (P.6),
besides its extensional implications, are truly misgivings about
matters of composition. We shall accordingly postpone their discussion
to Section 4, where we shall attend to these matters more fully. For
the moment, let us simply say that (P.6) is, on the face of it, not a
principle that can be added to M without further
argument.

One last important family of decomposition principles concerns the
question of atomism. Mereologically, an atom (or “simple”)
is an entity with no proper parts, regardless of whether it is
point-like or has spatial (and/or temporal) extension:

(32)

Atom
Ax =df ¬∃yPPyx.

By definition of ‘PP’, all atoms are pairwise disjoint and
can only overlap things of which they are part. Are there any such
entities? And, if there are, is everything entirely made up of atoms?
Is everything comprised of at least some atoms? Or is everything made up
of atomless “gunk”—as Lewis (1991: 20) calls
it—that divides forever into smaller and smaller parts? These
are deep and difficult questions, which have been the focus of
philosophical investigation since the early days of philosophy and
throughout the medieval and modern debate on anti-divisibilism, up to
Kant's antinomies in the Critique of Pure Reason (see the
entries on ancient atomism
and atomism from the 17th to the 20th
century). Along with nuclear physics, they made their way
into contemporary mereology mainly through Nicod's (1924)
“geometry of the sensible world”, Tarski's (1929)
“geometry of solids”, and Whitehead's (1929) theory of
“extensive connection” mentioned in Section 3.1, and are
now center stage in many mereological disputes at the intersection
between metaphysics and the philosophy of space and time (see, for
example, Sider 1993, Forrest 1996a, Zimmerman 1996, Markosian 1998a,
Schaffer 2003, McDaniel 2006, Hudson 2007a, Arntzenius 2008, and
J. Russell 2008, and the papers collected in Hudson 2004; see also
Sobociński 1971 and Eberle 1967 for some early treatments of
these questions in the spirit of Leśniewski's Mereology
and of Leonard and Goodman's Calculus of Individuals,
respectively). Here we shall confine ourselves to a brief
examination.

The two main options, to the effect that everything is ultimately made
up of atoms, or that there are no atoms at all, are typically
expressed by the following postulates, respectively:

(P.7)

Atomicity
∃y(Ay ∧ Pyx)

(P.8)

Atomlessness
∃yPPyx.

(See e.g. Simons 1987: 42.) These postulates are mutually
incompatible, but taken in isolation they can consistently be added to
any standard mereological theory X considered
here. Adding (P.7) yields a corresponding Atomistic version,
AX; adding (P.8) yields an Atomless version,
ÃX. Since finitude together with the
antisymmetry of parthood (P.3) jointly imply that mereological
decomposition must eventually come to an end, it is clear that any
finite model of M—and a fortiori of
any extension of M—must be
atomistic. Accordingly, an atomless mereology
ÃX admits only models of infinite
cardinality. An example of such a model, establishing the consistency
of the atomless versions of most standard mereologies considered in
this survey, is provided by the regular open sets of a Euclidean
space, with ‘P’ interpreted as set-inclusion (Tarski
1935). On the other hand, the consistency of an atomistic theory is
typically guaranteed by the trivial one-element model (with
‘P’ interpreted as identity), though one can also have
models of atomistic theories that allow for infinite domains. A case
in point is provided by the closed intervals on the real line, or the
closed sets of a Euclidean space (Eberle 1970). In fact, it turns out
that even when X is as strong as the full calculus of
individuals, corresponding to the theory GEM of
Section 4.4, there is no purely mereological formula that
says whether there are finitely or infinitely many atoms, i.e., that
is true in every finite model of AX but in no
infinite model (Hodges and Lewis 1968).

Concerning Atomicity, it is also worth noting that (P.7) does not
quite say that everything is ultimately made up of atoms; it
merely says that everything has atomic
parts.[16]
As such it rules out
gunky worlds, but one may wonder whether it fully captures the
atomistic intuition. In a way, the answer is in the affirmative. For,
assuming Reflexivity and Transitivity, (P.7) is equivalent to the
following

(33)

Pzx → ∃y(Ay ∧ Pyx ∧ Oyz),

which is logically equivalent to

(34)

((Ay ∧ Pyx) → Pyx) ∧
(Pzx → ∃y(Ay ∧ Pyx ∧
Oyz))

(adding a tautological conjunct), which is an instance of the general schema

(35)

(φy → Pyx) ∧ (Pzx →
∃y(φy ∧ Oyz)).

And (35) is the closest we can get to saying that x is
composed of the φs, i.e., all and only those entities that satisfy
the given condition φ (in the present case: being an atomic part
of x): every φ is part of x, and any part of
x overlaps some φ. Indeed, provided the φs are
pairwise disjoint, this is the standard definition of what it means
for something x to be composed of the φs (van Inwagen
1990: 29), and surely enough, if the φs are all atomic, then they
are pairwise disjoint. Thus, although (P.7) does not say that
everything is ultimately composed of atoms, it implies it—at
least in the presence of (P.1) and (P.2). (Of course, non-standard
mereologies in which either postulates is rejected may not warrant the
initial equivalence, so in such theories (33) would perhaps be a
better way to express the assumption of atomism.) In another way,
however, (34) may still not be enough. For if the domain is infinite,
(P.7) admits of models that seem to run afoul of the atomistic
doctrine. A simple example is a descending chain of decomposition that
never “bottoms out”, as in Figure 6: here x is
ultimately composed of atoms, but the pattern of decomposition that
goes down the right branch “looks” awfully similar to a
gunky precipice. For a concrete example (from Eberle 1970: 75),
consider the set of all subsets of the natural numbers, with parthood
modeled by the subset relation. In such a universe, each singleton
{n} will count as an atom and each infinite set {m:
m > n} will be “made up” of atoms. Yet
the set of all such infinite sets will be infinitely
descending. Models of this sort do not violate the idea that
everything is ultimately composed of atoms. However, they violate the
idea that everything can be decomposed into its ultimate
constituents. And this may be found problematic if atomism is meant to
carry the weight of metaphysical grounding: as J. Schaffer puts it,
the atomist's ontology seems to drain away “down a bottomless
pit” (2007: 184); being is “infinitely deferred, never
achieved” (2010: 62). Are there any ways available to the
atomist to avoid this charge? One option would simply be to require
that every model be finite, or that it involve only a finite set of
atoms. Yet such requirements, besides being philosophically harsh and
controversial even among atomists, cannot be formally implemented in
first-order mereology, the former for well-known model-theoretic
reasons and the latter in view of the above-mentioned result by Hodges
and Lewis (1968). The only reasonable option would seem to be a
genuine strengthening of Atomicity in the spirit of what Cotnoir
(2013c) calls “superatomism”. Given any object x,
(P.7) guarantees the existence of some parthood chain that
bottoms out at an atom. Superatomicity would require
that every parthood chain of x bottoms out—a
property that fails in the model of Figure 6. At the moment, such ways
of strengthening (P.7) have not been explored. However, in view of the
connection between classical mereology and Boolean algebras (see
below, Section 4.4), mathematical models for superatomistic
mereologies may be recovered from the work on superatomic Boolean
algebras initiated by Mostowski and Tarski (1939) and eventually
systematized in Day (1960). (A Boolean algebra is superatomic if and
only if every subalgebra is atomic, as with the algebra generated by
the finite subsets of a given set; see Day 1967 for an overview.) See
also Shiver (2015) for ways of strengthening (P.7) in the context of
stronger mereologies such as GEM (Section 4.4), or
within theories formulated in languages enriched with set variables or
plural quantification.

Another thing to notice is that, independently of their philosophical
motivations and formal limitations, atomistic mereologies admit of
significant simplifications in the axioms. For instance,
AEM can be simplified by replacing (P.5) and (P.7)
with

(P.5′)

Atomistic Supplementation
¬Pxy → ∃z(Az ∧
Pzx ∧ ¬Pzy),

which in turns implies the following atomistic variant of the
extensionality thesis (27):

(27′)

x=y ↔ ∀z(Az →
(Pzx ↔ Pzy)).

Thus, any atomistic extensional mereology is truly “hyperextensional” in
Goodman's (1958) sense: things built up from exactly the same atoms
are identical. In particular, if the domain of an
AEM-model has only finitely many atoms, the domain
itself is bound to be finite. An interesting question, discussed at
some length in the late 1960's (Yoes 1967, Eberle 1968, Schuldenfrei
1969) and taken up more recently by Simons (1987: 44f) and Engel and
Yoes (1996), is whether there are atomless analogues of
(27′). Is there any predicate that can play the role of
‘A’ in an atomless mereology? Such a predicate would
identify the “base” (in the topological sense) of the
system and would therefore enable mereology to cash out Goodman's
hyperextensional intuitions even in the absence of atoms. The question
is therefore significant especially from a nominalistic perspective,
but it has deep ramifications also in other fields (e.g., in
connection with a Whiteheadian conception of space according to which
space itself contains no parts of lower dimensions such as points or
boundary elements; see Forrest 1996a, Roeper 1997, and Cohn and Varzi
2003). In special cases there is no difficulty in providing a positive
answer. For example, in the ÃEM model
consisting of the open regular subsets of the real line, the open
intervals with rational end points form a base in the relevant
sense. It is unclear, however, whether a general answer can be given
that applies to any sort of domain. If not, then the only option would
appear to be an account where the notion of a “base” is
relativized to entities of a given sort. In Simons's terminology, we
could say that the ψ-ers form a base for the φ-ers if and only
if the following variants of (P.5′) and (P.7) are satisfied:

(P.5φ/ψ)

Relative Supplementation
(φx ∧ φy) → (¬Pxy
→ ∃z(ψz ∧ Pzx ∧
¬Pzy))

(P.7φ/ψ)

Relative Atomicity
φx → ∃y(ψy ∧ Pyx).

An atomistic mereology would then correspond to the limit case where
‘ψ’ is identified with the predicate ‘A’
for every choice of ‘φ’. In an atomless mereology, by
contrast, the choice of the base would depend each time on the level
of “granularity” set by the relevant specification of
‘φ’.

Concerning atomless mereologies, one more remark is in order. For
just as (P.7) is too weak to rule out unpleasant atomistic models, so
too the formulation of (P.8) may be found too weak to capture the
intended idea of a gunky world. For one thing, as it stands (P.8)
presupposes Antisymmetry. Absent (P.3), the symmetric two-element
pattern in Figure 3, left, would qualify as atomless. To rule out such
models independently of (P.3), one should understand (P.8) in terms of
the stronger notion of ‘PP’ given in (20′), i.e.,

(P.8′)

Proper Atomlessness
∃y(Pyx ∧ ¬Pxy).

Likewise, note that the pattern in Figure 2, middle, will qualify as a
model of (P.8) unless Supplementation is assumed, though again such a
pattern does not quite correspond to what philosophers ordinarily have
in mind when they talk about gunk. It is indeed an interesting
question whether Supplementation (or perhaps Quasi-supplementation, as
suggested by Gilmore 2015) is in some sense presupposed by the
ordinary concept of gunk. To the extent that it is, however, then
again one may want to be explicit, in which case the relevant
axiomatization may be simplified. For instance,
ÃMM can be simplified by merging (P.4) and
(P.8) into a single axiom:

(P.4′′)

Atomless Supplementation Pxy →
∃z(PPzy ∧ (Ozx
→ x=y)).

There is, in addition, another, more important sense in which (P.8)
may seem too week. After all, infinite divisibility is loose
talk. Given (P.8) (and also given (P.8′)), gunk may have
denumerably many, possibly continuum-many parts; but can it have more?
Is there an upper bound on the cardinality on the number of pieces of
gunk? Should it be allowed that for every cardinal number
there may be more than that many pieces of gunk? (P.8) is silent on
these questions. Yet these are certainly aspects of atomless mereology
that deserve scrutiny. It may even be thought that the world is not
mere gunk but “hypergunk”, as Nolan (2004: 305) calls
it—gunk such that, for any set of its parts, there is a set of
strictly greater cardinality containing only its parts. It is not
known whether such a theory is consistent (though Nolan conjectured
that a model can be constructed using the resources of standard set
theory with Choice and urelements together with some inaccessible
cardinal axioms), and even if it were, some philosophers would
presumably be inclined to regard hypergunk as a mere logical
possibility (Hazen 2004). Nonetheless the question is indicative of
the sort of leeway that (P.8) leaves, and that one might want to
regiment.

So much for the two main options, corresponding to atomicity and
atomlessness. What about theories that lie somewhere between these two
extremes? Surely it may be held that there are atoms, though not
everything need be made up of atoms; or it may be
held that there is atomless gunk, though not everything need be
gunky. (The latter position is defended e.g. by Zimmerman 1996.)
Formally, these possibilities can be put again in terms of suitable
restrictions on (P.7) and (P.8), by requiring that the relevant
conditions hold exclusively of certain entities:

(P.7φ)

φ-Atomicity
φx → ∀y(Pyx →
∃z(Az ∧ Pzy))

(P.8φ)

φ-Atomlessness
φx → ∀y(Pyx →
∃zPPzy).

And the options in question would correspond to endorsing
(P.7φ) or (P.8φ) for specific values of
‘φ’. At present, no thorough formal investigation has
been pursued in this spirit (though see Masolo and Vieu 1999 and
Hudson 2007b). Yet the issue is particularly pressing when it comes to
the mereology of the spatio-temporal world. For example, it is a
plausible thought that while the question of atomism may be left open
with regard to the mereological structure of material objects (pending
empirical findings from physics), one might be able to settle it
(independently) with regard to the structure of space-time
itself. This would amount to endorsing a version of either
(P.7φ) or (P.8φ) in which
‘φ’ is understood as a condition that is satisfied
exclusively by regions of space-time. Some may find it hard to
conceive of a world in which an atomistic space-time is inhabited by
entities that can be decomposed indefinitely (pace McDaniel 2006), in which case accepting
(P.7φ) for regions would entail the stronger principle
(P.7). However, (P.8φ) would be genuinely independent
of (P.8) unless it is assumed that every mereologically atomic entity
should be spatially unextended, an assumption that is not part of
definition (32) and that has been challenged by van Inwagen (1981) and
Lewis (1991: 32) (and extensively discussed in recent literature; see
e.g. MacBride 1998, Markosian 1998a, Scala 2002, J. Parsons 2004,
Simons 2004, Tognazzini 2006, Braddon-Mitchell and Miller 2006, Hudson
2006a, McDaniel 2007, Sider 2007, Spencer 2010). More generally, such
issues depend on the broader question of whether the mereological
structure of a thing should always “mirror” or be in
perfect “harmony” with that of its spatial or
spatio-temporal receptacle, a question addressed in J. Parsons (2007)
and Varzi (2007: §3.3) and further discussed in Schaffer (2009),
Uzquiano (2011) and Saucedo (2011). (For more on this, see the entry
location and mereology.)

Similar considerations apply to other decomposition principles that
may come to mind at this point. For example, one may consider a
requirement to the effect that ‘PP’ forms a dense
ordering, as already Whitehead (1919) had it:

(P.9)

Density
PPxy → ∃z(PPxz ∧ PPzy).

As a general decomposition principle, (P.9) might be deemed too
strong, especially in an atomistic setting. (Whitehead's own theory
assumes Atomlessness.) However, it is plausible to suppose that (P.9)
should hold at least with respect to the domain of spatio-temporal
regions, regardless of whether these are construed as atomless gunk or
as aggregates of spatio-temporal atoms. For more on this, see
Eschenbach and Heydrich (1995) and Varzi (2007: §3.2).

Finally, it is worth noting that if one assumed the existence of a
“null item” that is part of everything, corresponding to
the postulate

(P.10)

Bottom
∃x∀yPxy,

then such an entity would perforce be an atom. Accordingly, no
atomless mereology is compatible with this assumption. But it bears
emphasis that (P.10) is at odds with a host of other theories as
well. For, given (P.10), the Antisymmetry axiom (P.3) will immediately
entail that the atom in question is unique, while the Reflexivity
axiom (P.1) will entail that it overlaps everything, hence that
everything overlaps everything. This means that under such axioms the
Supplementation principle (P.4) cannot be satisfied except in models
whose domain includes a single element. Indeed, this is also true of
the weaker Quasi-supplementation principle, (P.4d). It
follows, therefore, that the result of adding (P.10) to any theory at
least as strong as (P.1) + (P.3) + (P.4d), and a fortiori
to MM and any extension thereof, will immediately
collapse to triviality in view of the following corollary:

(36)

∃x∀y x=y.

‘Triviality’ may strike one as the wrong word here. After
all, there have been and continue to be philosophers who hold
radically monistic ontologies—from the Eleatics (Rea 2001) to
Spinoza (J. Bennett 1984) all the way to contemporary authors such as
Horgan and Potrč (2000), whose comparative ontological parsimony
results in the thesis that the whole cosmos is but one huge extended
atom, an enormously complex but partless “blobject”. For
all we know, it may even be that the best ontology for quantum
mechanics, if not for Newtonian mechanics, consists in a lonely atom
speeding through configuration-space (Albert 1996). None of this is
trivial. However, none of this corresponds to fully endorsing (36),
either. For such philosophical theories do not, strictly speaking,
assert the existence of one single entity—which is what
(36) says—but only the existence of a single material substance
along with entities of other kinds, such as properties or
spatio-temporal regions. In other words, they only endorse a sortally
restricted version of (36). In its full generality, (36) is much
stronger and harder to swallow, and most mereologists would rather
avoid it. The bottom line, therefore, is that theories endorsing
(P.10) are likely to be highly non-standard, pace Carnap's
persuasion that the null item would be a “natural and convenient
choice” for certain purposes (such as providing a referent for
all defective descriptions; see 1947: 37). A few authors have indeed
gone that way, beginning with Martin (1943, 1965), who rejects
unrestricted Reflexivity and characterizes the null item as
“that which is not part of itself”. Other notable
exceptions include Bunt (1985) and Meixner (1997) and, more recently,
Hudson (2006) and Segal (2014), both of whom express sympathy for the
null individual at the cost of foregoing unrestricted
(Quasi-)Supplementation. See also Priest (2014: §6.13) and
Cotnoir and Weber (2014), who avoid (36) through a paraconsistent
recasting of the underlying logic. Still another option would be to
treat the null item as a mere algebraic “fiction” and to
amend the entire mereological machinery accordingly, carefully
distinguishing between trivial cases of parthood and overlap (those
that involve the infectious null item) and genuine, non-trivial
ones:

(37)

Genuine Parthood
GPxy =df Pxy ∧
∃z¬Pxz

(38)

Genuine Overlap GOxy =df
∃z(GPzx ∧ GPzy).

The basic M-axioms need not be affected by this
distinction. But stronger principles such as Supplementation could
give way to their “genuine” counterparts, as in

(P.4G)

Genuine Supplementation
PPxy → ∃z(GPzy ∧ ¬GOzx),

and this would suffice to block the inference to (36) while keeping
with the spirit of standard mereology. This strategy is not uncommon,
especially in the mathematically oriented literature (see e.g. Mormann
2000, Forrest 2002, Pontow and Schubert 2006), and we shall briefly
return to it in Section 4.4 below. In general, however, mereologists
tend to side with traditional wisdom and steer clear of (P.10)
altogether.

Let us now consider the second way of extending M
mentioned at the beginning of Section 3. Just as we may want to
regiment the behavior of P by means of decomposition principles that
take us from a whole to its parts, we may look at composition
principles that go in the opposite direction—from the parts to
the whole. More generally, we may consider the idea that the domain of
the theory ought to be closed under mereological operations of various
sorts: not only mereological sums, but also products, differences, and
more.

Conditions on composition are many. Beginning with the weakest, one
may consider a principle to the effect that any pair of suitably
related entities must underlap, i.e., have an upper bound:

(P.11ξ)

ξ-Bound
ξxy → ∃z(Pxz ∧ Pyz).

Exactly how ‘ξ’ should be construed is, of course,
an important question by itself—a version of what van Inwagen
(1987, 1990) calls the “Special Composition Question”. A
natural choice would be to identify ξ with mereological overlap,
the rationale being that such a relation establishes an important tie
between what may count as two distinct parts of a larger whole. As we
shall see (Section 4.5), with ξ so construed (P.11ξ)
is indeed rather uncontroversial. By contrast, the most liberal choice
would be to identify ξ with the universal relation, in which case
(P.11ξ) would reduce to its consequent and assert the
existence of an upper bound for any pair of entities
x and y. An axiom of this sort was used, for
instance, in Whitehead's (1919, 1920) mereology of
events.[17]
In any case, and
regardless of any specific choice, it is apparent that
(P.11ξ) does not express a strong condition on
composition, as the consequent is trivially satisfied in any domain
that includes a universal entity of which everything is part, or any
entity sufficiently large to include both x and y as
parts regardless of how they are related.

A stronger condition would be to require that any pair of
suitably related entities must have a minimal
underlapper—something composed exactly of their parts and
nothing else. This requirement is sometimes stated by saying that any
suitable pair must have a mereological “sum”, or
“fusion”,[18]
though it is not immediately obvious how this
requirement should be formulated. Consider the following definitions:

(‘Sizxy’ may be read:
‘z is a sumi of x and
y’. The first notion is found e.g. in Eberle 1967,
Bostock 1979, and van Benthem 1983; the second in Tarski 1935 and
Lewis 1991; the third in Needham 1981, Simons 1987, and Casati and
Varzi 1999.) Then, for each i ∈ {1, 2, 3}, one could
extend M by adding a corresponding axiom as follows,
where again ξ specifies a suitable binary condition:

(P.12ξ,i)

ξ-Sumi
ξxy → ∃zSizxy.

In a way, (P.12ξ,1) would seem the obvious choice,
corresponding to the idea that a sum of two objects is just a minimal
upper bound of those objects relative to P (a partial
ordering). However, this condition may be regarded as too weak to
capture the intended notion of a mereological sum. For example, with
ξ construed as overlap, (P.12ξ,1) is satisfied by the
model of Figure 7, left: here z is a minimal upper bound
of x and y, yet z hardly qualifies as a sum
“made up” of x and y, since its parts
include also a third, disjoint item w. Indeed, it is a simple
fact about partial orderings that among finite models
(P.12ξ,1) is equivalent to (P.11ξ), hence
just as weak.

By contrast, (P.12ξ,2) corresponds to a notion of sum
that may seem too strong. In a way, it says—literally—that
any pair of suitably ξ-related entities x and y
compose something, in the sense already discussed in connection with
(35): they have an upper bound all parts of which overlap either
x or y. Thus, it rules out the model on the left of
Figure 7, precisely because w is disjoint from both
x and y. However, it also rules out the model on the
right, which depicts a situation in which z may be viewed as
an entity truly made up of x and y insofar as it is
ultimately composed of atoms to be found either in x or in
y. Of course, such a situation violates the Strong
Supplementation principle (P.5), but that's precisely the sense in
which (P.12ξ,2) may seem too strong: an
anti-extensionalist might want to have a notion of sum that does not
presuppose Strong Supplementation.

The formulation in (P.12ξ,3) is the natural
compromise. Informally, it says that for any pair of suitably
ξ-related entities x and y there is something
that overlaps exactly those things that overlap either x or
y. This is strong enough to rule out the model on the left,
but weak enough to be compatible with the model on the right. Note,
however, that if the Strong Supplementation axiom (P.5) holds, then
(P.12ξ,3) is equivalent to
(P.12ξ,2). Moreover, it turns out that if the stronger
Complementation axiom (P.6) holds, then all of these principles are
trivially satisfied in any domain in which there is a universal
entity: in that case, regardless of ξ, the sum of any two entities
is just the complement of the difference between the complement of one
minus the other. (Such is the strength of (P.6), a genuine cross
between decomposition and composition principles.)

Figure 7. A sum1 that is not a
sum3, and a sum3 that is not a
sum2.

The intuitive idea behind these principles is in fact best appreciated
in the presence of (P.5), hence extensionality, for in that case the
relevant sums must be unique. Thus, consider the following definition,
where i ∈ {1, 2, 3} and ‘℩’ is the
definite descriptor):

(40i)

x +i y
=df ℩zSizxy.

In the context of EM, each (P.12ξ,i)
would then imply that the corresponding sum operator has all the
“Boolean” properties one might
expect (Breitkopf 1978). For example, as long as the arguments satisfy
the relevant condition
ξ,[20] each
+i is idempotent, commutative, and associative,

(41)

x = x +i x

(42)

x +i y = y +i x

(43)

x +i (y +i z) =
(x +i y) +i z,

and well-behaved with respect to parthood:

(44)

Px(x +i y)

(45)

Pxy → Px(y +i z)

(46)

P(x +i y)z → Pxz

(47)

Pxy ↔ x +i y = y.

(Note that (47) would warrant defining ‘P’ in terms of
‘+i’, treated as a primitive. For
i=3, this was actually the option endorsed in Leonard 1930:
187ff.)

Indeed, here there is room for further developments. For example, just
as the principles in (P.12ξ,i) assert the
existence of a minimal underlapper for any pair of suitably related
entities, one may at this point want to assert the existence of a
maximal overlapper, i.e., not a “sum” but a
“product” of those entities. In the present context, such
an additional claim can be expressed by the following principle:

(P.13ξ)

ξ-Product ξxy →
∃zRzxy,

where

(48)

Product
Rzxy =df ∀w(Pwz ↔
(Pwx ∧ Pwy)),

and ‘ξ’ is at least as strong as ‘O’
(unless one assumes the Bottom principle
(P.10)). In EM one could then introduce the
corresponding binary operator,

(49)

x × y =df
℩zRzxy,

and it turns out that, again, such an operator would have the
properties one might expect. For example, as long as the arguments
satisfy the relevant condition ξ, × is idempotent,
commutative, and associative, and it interacts with each
+i in conformity with the usual absorption
laws:

(50)

x +i (y × z)
= (x +iy) × (x
+iz)

(51)

x × (y +iz)
= (x × y) +i (x
× z).

Now, obviously (P.13ξ) does not qualify as a composition
principle in the main sense that we have been considering here, i.e.,
as a principle that yields a whole out of suitably ξ-related
parts. Still, in a derivative sense it does. It asserts the existence
of a whole composed of parts that are shared by suitably
related entities. Be that as it may, it should be noted that such an
additional principle is not innocuous unless ‘ξ’
expresses a condition stronger than mere overlap. For instance, we
have said that overlap may be a natural option if one is unwilling to
countenance arbitrary scattered sums. It would not, however, be enough
to avoid embracing scattered products. Think of two C-shaped objects
overlapping at both extremities; their sum would be a one-piece
O-shaped object, but their product would consist of two disjoint,
separate parts (Bostock 1979: 125). Moreover, and independently, if
ξ were just overlap, then (P.13ξ) would be
unacceptable for anyone unwilling to embrace mereological
extensionality. For it turns out that the Strong Supplementation
principle (P.5) would then be derivable from the weaker
Supplementation principle (P.4) using only the partial ordering axioms
for ‘P’ (in fact, using only Reflexivity and Transitivity;
see Simons 1987: 30f). In other words, unless ‘ξ’
expresses a condition stronger than overlap, MMcum (P.13ξ) would automatically
include EM. This is perhaps even more remarkable, for
on first thought the existence of products would seem to have nothing
to do with matters of decomposition, let alone a
decomposition principle that is committed to extensionality. On second
thought, however, mereological extensionality is really a
double-barreled thesis: it says that two wholes cannot be decomposed
into the same proper parts but also, by the same token, that two
wholes cannot be composed out of the same proper parts. So it is not
entirely surprising that as long as proper parthood is well behaved,
as per (P.4), extensionality might pop up like this in the presence of
substantive composition principles. (It is, however, noteworthy that
it already pops up as soon as (P.4) is combined with a seemingly
innocent thesis such as the existence of products, so the
anti-extensionalist should keep that in mind.)

One can get even stronger composition principles by considering
infinitary bounds and sums. For example, (P.11ξ) can be
generalized to a principle to the effect that any non-empty
set of (two or more) entities satisfying a suitable condition
ψ has an upper bound. Strictly speaking, there is a difficulty in
expressing such a principle in a standard first-order language. Some
early theories, such as those of Tarski (1929) and Leonard and Goodman
(1940), require explicit quantification over sets (see Niebergall
2009a, 2009b; Goodman produced a set-free version of the calculus of
individuals in 1951). Others, such as Lewis's (1991), resort to the
machinery of plural quantification of Boolos (1984). One can, however,
avoid all this and achieve a sufficient degree of generality by
relying on an axiom schema where sets are identified by predicates or
open formulas. Since an ordinary first-order language has a
denumerable supply of open formulas, at most denumerably many sets (in
any given domain) can be specified in this way. But for most purposes
this limitation is negligible, as normally we are only interested in
those sets of objects that we are able to specify. Thus, for most
purposes the following axiom schema will do, where ‘φ’
is any formula in the language and ‘ψ’ expresses the
condition in question:

(P.14ψ)

General ψ-Bound
(∃wφw ∧
∀w(φw → ψw)) →
∃z∀w(φw →
Pwz).

(The first conjunct in the antecedent is simply to guarantee that
‘φ’ picks out a non-empty set, while in the
consequent the variable ‘z’ is assumed not to
occur free in ‘ψ’.) The three binary sum axioms
corresponding to the schema in (P.12ξ,i) can be
strengthened in a similar fashion as follows:

(Here, ‘Sizφw’ may be
read: ‘z is a sumi of
every w such that φw’ and, again,
‘z’ and ‘v’ are assumed not
to occur free in φ; similar restrictions will apply below.) Thus,
each (P.15ψ,i) says that if there are some
φ-ers, and if every φ-er satisfies condition ψ, then the
φ-ers have a sum of the relevant type. It can be checked that each
variant of (P.15ψ,i) includes the
corresponding finitely principle (P.12ψ,i) as
a special case, taking ‘φw’ to be the formula
‘w=x ∨ w=y’ and
‘ψw’ the condition
‘(w=x → ξwy) ∧
(w=y → ξxw)’. And, again, it
turns out that in the presence of Strong Supplementation,
(P.15ψ,2) and (P.15ψ,3) are
equivalent.

One could also consider here a generalized version of the Product
principle (P.13ξ), asserting the conditional existence
of a maximal common overlapper—a common “nucleus”,
in the terminology of Leonard and Goodman (1940)—for any
non-empty set of entities satisfying a suitable condition. Adapting
from Goodman (1951: 37), such a principle could be stated as
follows:

(P.16ψ)

General ψ-Product
(∃wφw ∧
∀w(φw → ψw)) →
∃zRzφw,

where

(53)

General Product
Rzφw =df ∀v(Pvz ↔
∀w(φw → Pvw))

and ‘ψw’ expresses a condition at
least as strong as ‘∀x(φx →
Owx)’ (again, unless one assumes the Bottom principle
(P.10)). This principle includes the finitary version
(P.13ξ) as a special case, taking
‘φw’ and ‘ψw’ as
above, so the remarks we made in connection with the latter apply
here. An additional remark, however, is in order. For there is a sense
in which (P.16ψ) might be thought to be redundant in the
presence of the infinitary sum principles in (P.15ψ,i).
Intuitively, a maximal common overlapper (i.e., a
product) of a set of overlapping entities is simply a minimal
underlapper (a sum) of their common parts; that is precisely the
sense in which a product principle qualifies as a composition
principle. Thus, intuitively, each of the infinitary sum principles
above should have a substitution instance that yields
(P.16ψ) as a theorem, at least when
‘ψw’ is as strong as indicated. However, it
turns out that this is not generally the case unless one assumes
extensionality. In particular, it is easy to see that
(P.15ψ,3) does not generally imply
(P.16ψ), for it may not even imply the binary version
(P.13ξ). This can be verified by taking
‘ξxy’ and ‘ψw’ to
express just the requirement of overlap, i.e., the conditions
‘Oxy’ and
‘∀x(φx → Owx)’,
respectively, and considering again the non-extensional model
diagrammed in Figure 4. In that model, x and y do
not have a product, since neither is part of the other and neither z
nor w includes the other as a part. Thus,
(P.13ξ) fails, which is to say that
(P.16ψ) fails when ‘φ’ picks out the
set {x, y}; yet (P.15ψ,3) holds, for
both z and w are things that
overlap exactly those things that overlap some common part of the
φ-ers, i.e., of x and y.

In the literature, this fact has been neglected until recently (Pontow
2004). It is, nonetheless, of major significance for a full
understanding of (the limits of) non-extensional mereologies. As we
shall see in the next section, it is also important when it comes to
the axiomatic structure of mereology, including the axiomatics of the
most classical theories.

The strongest versions of all these composition principles are
obtained by asserting them as axiom schemas holding for every
condition ψ, i.e., effectively, by foregoing any reference to
ψ altogether. Formally this amounts in each case to dropping the
second conjunct of the antecedent, i.e., to asserting the schema
expressed by the relevant consequent with the only proviso that there
are some φ-ers. In particular, the following schema is the
unrestricted version of (P.15ψ,i), to the
effect that every specifiable non-empty set of entities has a
sumi:

(P.15i)

Unrestricted Sumi
∃wφw → ∃zSizφw.

For i=3, the extension of EM obtained by adding every instance
of this schema has a distinguished pedigree and is known in the
literature as General Extensional Mereology, or
GEM. It corresponds to the classical systems of
Leśniewski and of Leonard and Goodman, modulo the underlying
logic and choice of primitives. The same theory can be obtained by
extending EM with (P.152) instead, for in
the presence of extensionality the two schemas are equivalent. Indeed,
it turns out that the latter axiomatization is somewhat redundant:
given just Transitivity and Supplementation, Unrestricted
Sum2 entails all the other axioms,
i.e., GEM is the same theory as (P.2) + (P.4) +
(P.152). By contrast, extending EM with
(P.151) would result in a weaker theory (Figure 8), though
one can still get the full strength of GEM with the
help of additional axioms. For example, Hovda (2009) shows that the
following will do:

(P.17)

Filtration (S1zφw
∧ Pxz) → ∃w(φw ∧
Owx).

(in which case, again, Transitivity and Supplementation would suffice,
i.e., GEM = (P.2) + (P.4) + (P.151) +
(P.17)). For other ways of axiomatizatizing of GEM
using (P.151), see e.g. Link (1983) and Landman (1991)
(and, again, Hovda 2009). See also Sharvy (1980, 1983), where the
extension of M obtained by adding (P.151)
is called a “quasi-mereology”.

Figure 8. A model of EM +
(P.152) but not of GEM.

GEM is a powerful theory, and it was meant to be so
by its nominalistic forerunners, who were thinking of mereology as a
good alternative to set theory. It is also decidable (Tsai 2013a),
whereas for example, M, MM, and
EM, and many extensions thereof turn out to be
undecidable. (For a comprehensive picture of decidability in
mereology, see also Tsai 2009, 2011, 2013b.) Just how powerful is
GEM? To answer this question, let us focus on the
classical formulation based on (P.153) and consider the
following generalized sum operator:

(54)

General Sum σxφx
=df ℩zS3zφw.

Then (P.153) and (P.5) can be simplified to a single axiom schema:

(P.18)

Unique Unrestricted Sum3
∃xφx →
∃z(z=σxφx),

and we can introduce the following definitions:

(55)

Sumx + y =df σz(Pzx
∨
Pzy)

(56)

Productx × y =df σz(Pzx
∧ Pzy)

(57)

Differencex − y =df σz(Pzx ∧
Dzy)

(58)

Complement
~x =df σzDzx

(59)

UniverseU =df σzPzz.

Note that (55) and (56) yield the binary operators defined in (403)
and (49) as special cases. Moreover, in GEM the
General ψ-Product principle (P.16ψ) is also derivable
as a theorem, with ‘ψ’ as weak as the requirement of
mutual overlap, and we can introduce a corresponding functor as
follows:

(60)

General Product πxφx
=df σz∀x(φx
→ Pzx).

The full strength of the theory can then be appreciated by
considering that its models are closed under each of these functors,
modulo the satisfiability of the relevant conditions. To be explicit:
the condition ‘DzU’ is unsatisfiable,
so U cannot have a complement. Likewise products are defined
only for overlappers and differences only for pairs that leave a
remainder. Otherwise, however, (55)–(60) yield perfectly
well-behaved functors. Since such functors are the natural
mereological analogues of the familiar set-theoretic operators, with
‘σ’ in place of set abstraction, it follows that the
parthood relation axiomatized by GEM has essentially
the same properties as the inclusion relation in standard set
theory. More precisely, it is isomorphic to the inclusion relation
restricted to the set of all non-empty subsets of a given set, which
is to say a complete Boolean algebra with the zero element
removed—a result that can be traced back to Tarski (1935:
n. 4).[22]

There are other equivalent formulations of GEM that
are noteworthy. For instance, it is a theorem of every extensional
mereology that parthood amounts to inclusion of overlappers:

(61)

Pxy ↔ ∀z(Ozx → Ozy).

This means that in an extensional mereology ‘O’ could
be used as a primitive and ‘P’ defined accordingly, as in
Goodman (1951), and it can be checked that the theory defined by
postulating (61) together with the Unrestricted Sum principle (P.153)
and the Antisymmetry axiom (P.3) is equivalent to
GEM (Eberle 1967). Another elegant axiomatization
of GEM, due to an earlier work of Tarski
(1929),[23] is
obtained by taking just the Transitivity axiom (P.2) together with the
Sum2-analogue of the Unique Unrestricted Sum axiom
(P.18). By contrast, it bears emphasis that the result of adding
(P.153) to MM is not equivalent
to GEM, contrary to the “standard”
characterization given by Simons (1987: 37) and inherited by much
literature that followed, including Casati and Varzi (1999) and the first
edition of this entry.[24]
This follows immediately from Pontow's
(2004) counterexample mentioned at the end of Section 4.3, since the
non-extensional model in Figure 4 satisfies (P.153), and was first
noted in Pietruszczak (2000, n. 12). More generally, in Section 4.2 we
have mentioned that in the presence of the binary Product postulate
(P.13ξ), with ξ construed as overlap, the Strong
Supplementation axiom (P.5) follows from the weaker Supplementation
axiom (P.4). However, the model shows that the postulate is not
implied by (P.153) any more than it is implied by its restricted
variants (P.15ψ,3). Apart from its relevance to the
proper characterization of GEM, this result is worth
stressing also philosophically, for it means that (P.153) is by itself
too weak to generate a sum out of any specifiable set of objects. In
other words, fully unrestricted composition calls for extensionality,
on pain of giving up both supplementation principles. The
anti-extensionalist should therefore keep that in mind. (On the other
hand, a friend of extensionality may welcome this result as an
argument in favor of adopting (P.152) instead of
(P.153), for we have already noted that such a way of
sanctioning unrestricted composition turns out to be enough,
in MM, to entail Strong Supplementation along with
the existence of all products and, with them, of all sums; see Varzi
2009, with discussion in Rea 2010 and Cotnoir 2015a . In this sense,
the standard way of characterizing composition given in (35), on which
(P.152) is based, is not as neutral as it might seem. On
this and related matters, indicating that the axiomatic path to
“classical extensional mereology” is no straightforward
business, see also Hovda 2009 and Gruszczyński and Pietruszczak
2014.)

Would we get a full Boolean algebra by supplementing
GEM with the Bottom axiom (P.10), i.e., by positing
the mereological equivalent of the empty set? One immediate way to
answer this question is in the affirmative, but only in a trivial
sense: we have already seen in Section 3.4 that, under the axioms of
MM, (P.10) only admits of degenerate one-element
models. Such is the might of the null item. On the other hand, suppose
we rely on the “non-trivial” notions of genuine parthood
and genuine overlap defined in (37)–(38). And suppose we
introduce a corresponding family of “non-trivial”
operators for sum, product, etc. Then it can be shown that the theory
obtained from GEM by adding (P.10) and replacing
(P.5) and (P.153) with the following non-trivial variants:

(P.5G)

Genuine Strong Supplementation
¬Pyx → ∃z(GPzy ∧ ¬GOzx)

(P.153G)

Genuine Unrestricted Sum3
∃wφw →
∃z∀v(GOzv ↔
∃w(φw ∧
GOwv))

is indeed a full Boolean algebra under the new operators (Pontow
and Schubert 2006). This shows that, mathematically, mereology does
indeed have all the resources to stand as a robust and yet
nominalistically acceptable alternative to set theory, the real source
of difference being the attitude towards the nature of singletons (as
already emphasized by Leśniewski 1916 and eventually clarified by
Lewis 1991). As already mentioned, however, from a philosophical
perspective the Bottom axiom is by no means a favorite option. The
null item would have to exist “nowhere and nowhen” (as
Geach 1949: 522 put it), or perhaps “everywhere and
everywhen” (as in Efird and Stoneham 2005), and that is hard to
swallow. One may try to justify the gulp in various ways, perhaps by
construing the null item as a non-existing individual (Bunge 1966), as
a Meinongian object lacking all nuclear properties (Giraud 2013), as
an Heideggerian nothing that nothings (Priest 2014: §6.13), or as
the ultimate incarnation of divine omnipresent simplicity (Hudson
2006b, 2009). But few philosophers would be willing to go ahead and
swallow for the sole purpose of neatening up the algebra.

Finally, it is worth recalling that the assumption of atomism
generally allows for significant simplifications in the axiomatics of
mereology. For instance, we have already seen that
AEM can be simplified by subsuming (P.5) and (P.7)
under a single Atomistic Supplementation principle,
(P.5′). Likewise, it is easy to see that GEM is
compatible with the assumption of Atomicity (just consider the
one-element model), and the resulting theory has some attractive
features. In particular, it turns out that AGEM can
be simplified by replacing any of the Unrestricted Sum postulates
in (P.15i) with the more perspicuous

(P.15i′)

Atomistic Sumi
∃wφw →
∃zSiz(Av ∧
∃w(φw ∧ Pvw)),

which asserts, for any non-empty set of entities, the existence of a
sumi composed exactly of all the atoms that
compose those entities. Indeed, GEM also provides
the resources to overcome the limits of the Atomicity axiom (P.7)
discussed in Section 3.4. For, on the one hand, the infinitely
descending chain depicted in Figure 6 is not a model of
AGEM, since it is missing all sorts of sums. On the
other, in GEM one can actually strenghten (P.7) in
such a way as to require explicitly that everything be made entirely
atoms, as in

(P.7′)

Strong Atomicity
∃yAy ∧
PxσyAy.

(See Shiver 2015.) It should be noted, however, that such advantages
come at a cost. For regardless of the number of atoms one begins with,
the axioms of AGEM impose a fixed relationship
between that number, κ, and the overall number of things, which
is going to be 2κ–1. As Simons (1987: 17)
pointed out, this means that the possible cardinality of an
AGEM-model is restricted. There are models with 1, 3,
7, 15, 2ℵ0, and many more cardinalities,
but no models with, say, cardinality 2, 4, 6, or
ℵ0. Obviously, this is not a consequence of
(P.15i) alone but also of the other axioms
of GEM (the unsupplemented pattern in Figure 2, left,
satisfies (P.15i) for each i and has 2
elements, and can be expanded at will to get models of any finite
cardinality, or indefinitely to get a model with ℵ0
elements, as in Figure 2, center; see also Figure 8 for a supplemented
non-filtrated model of (P.151) with 4 elements and Figure
7, right, for a supplemented non-extensional model of
(P.153) with 6 elements). Still, it is a fact that in the
presence of such axioms each (P.15i) rules out a
large number of possibilities. In particular, every finite model of
AGEM—hence of GEM—is
bound to involve massive violations of what Comesaña (2008)
calls “primitive cardinality”, namely, the intuive thesis
to the effect that, for any integer
n, there could be exactly n things. And since the
size of any atomistic domain can always be reached from below by
taking powers, it also follows that AGEM cannot have
infinite models of strongly inaccessible cardinality. Such is, as
Uzquiano (2006) calls it, the “price of universality” in
the context of Atomicity.

What about ÃGEM, the result of adding the
Atomlessness axiom (P.8)? Obviously the above limitation does not
apply, and the Tarski model mentioned in Section 3.4 will suffice to
establish consistency. However, note that every GEM
model—hence every ÃGEM model—is
necessarily bound at the top, owing to the existence of the universal
entity U. This is not by itself problematic: while the
existence of U is the dual the Bottom axiom, a top jumbo of
which everything is part has none of the formal and philosophical
oddities of a bottom atom that is part of everything (though see
Section 4.5 for qualifications). Yet a philosopher who believes in
infinite divisibility, or at least in its possibility, might feel the
same about infinite composability. Just as everything could be made of
atomless gunk that divides forever into smaller and smaller parts,
everything might be mereological “junk”—as Schaffer
(2010: 64) calls it—that composes forever into greater and
greater wholes. (One philosopher who held such a view is, again,
Whitehead, whose mereology of events includes both the Atomlessness
principle and its upward dual, i.e.:

(P.19)

Ascent
∃yPPxy.

See Whitehead 1919: 101; 1920: 76). GEM is compatible
with the former possibility, and ÃGEM makes it
into a universal necessity. But neither has room for the
latter. Indeed, the possibility of junk might be attractive also from
an atomist perspective. After all, already Theophilus thought that
even though everything is composed of monads, “there is never an
infinite whole in the world, though there are always wholes greater
than others ad infinitum” (Leibniz, New
Essays, I-xiii-21). Is this a serious limitation of
GEM? More generally, is this a serious limitation of
any theory in which the existence of U is a
theorem—effectively, any theory endorsing at least the
unrestricted version of (P.14ψ)? (In the absence of
Antisymmetry, one may want to consider this question by understanding
the predicate ‘PP’ in (P.19) in terms of the stronger
definition given in (20′); see above, ad (P.8′).)
Some authors have argued that it is (Bohn 2009a, 2009b, 2010), given
that junk is at least conceivable (see also Tallant 2013) and admits
of plausible cosmological and mathematical models (Morganti 2009,
Mormann 2014). Others have argued that it isn't, because junk is
metaphysically impossible (Schaffer 2010, Watson 2010). Others still
are openly dismissive about the question (Simons 1987: 83). One may
also take the issue to be symptomatic of the sorts of trouble that
affect any theory that involves quantification over absolutely
everything, as the Unrestricted Sum principles in
(P.15i) obviously do (see Spencer 2012, though his
remarks focus on mereological theories formulated in terms of plural
quantification). One way or the other, from a formal perspective the
incompatibility with Ascent may be viewed as an unpleasant consequence
of (P.15i), and a reason to go for weaker
theories. In particular, it may be viewed as a reason to endorse only
finitary sums, which is to say only instances of
(P.12ξ,i), or perhaps its unrestricted
version:

(P.12i)

Finitary Unrestricted Sumi
∃zSizxy.

(See Contessa 2012 and Bohn 2012: 216 for explicit suggestions in this
spirit.) This would be consistent with the existence of junky worlds
as it is consistent with the existence of gunky worlds. Yet it should
be noted that even this move has its costs. For example, it turns out
that in a world that is both gunky and junky (what Bohn calls
“hunk”) (P.12i) is in tension with the
Complementation principle (P.6) for each i (Cotnoir
2014). Moreover, while (P.12i) is compatible with
junky worlds, i.e., models that fully satisfy the Ascent
axiom (P.19), it is in tension with the possibility of worlds
containing junky structures along with other, disjoint elements
(Giberman 2015).

The algebraic strength of GEM, and of its weaker
finitary and infinitary variants, is worth emphasizing, but it also
reflects substantive mereological postulates whose philosophical
underpinnings leave room for considerable controversy well beyond the
gunk/junk dispute. Indeed, all composition principles turn
out to be controversial, just as the decomposition principles examined
in Section 3. For, on the one hand, it appears that the weaker,
restricted formulations, from (P.11ξ) to
(P.15ψ,i), are just not doing enough work: not only do
they depend on the specification of the relevant limiting conditions,
as expressed by the predicates ‘ξ’ and
‘ψ’; they also treat such conditions as merely
sufficient for the existence of bounds and sums, whereas ideally we
are interested in an account of conditions that are both sufficient
and necessary. On the other hand, the stronger, unrestricted
formulations appear to go too far, for while they rule out the
possibility of junky worlds, they also commit the theory to the
existence of a large variety of prima facie implausible,
unheard-of mereological composites—a large variety of
“junk” in the good old sense of the word.

Concerning the first sort of worry, one could of course construe every
restricted composition principle as a biconditional expressing both a sufficient
and a necessary condition for the existence of an upper bound,
or a sum, of a given pair or set of entities. But then the question of
how such conditions should be construed becomes crucial, on pain of
turning a weak sufficient condition into an exceedingly strong
requirement. For example, with regard to (P.11ξ) we
have mentioned the idea of construing ‘ξ’ as
‘O’, the rationale being that mereological overlap
establishes an important connection between what may count as two
distinct parts of a larger (integral) whole. However, as a necessary
condition overlap is obviously too stringent. The top half of my body
and the bottom half do not overlap, yet they do form an integral
whole. The topological relation of contact, i.e., overlap or
abut, might be a better candidate. Yet even that would be too stringent. We
may have misgivings about the existence of scattered entities
consisting of totally disconnected parts, such as my umbrella and your
left shoe or, worse, the head of this trout and the body of that
turkey (Lewis 1991: 7–8). Yet in other cases it appears
perfectly natural to countenance wholes that are composed of two or
more disconnected entities: a bikini, a token of the lowercase letter
‘i’, my copy of The Encyclopedia of Philosophy
(R. Cartwright 1975; Chisholm 1987)—indeed any garden-variety
material object, insofar as it turns out to be a swarm of spatially
isolated elementary particles (van Inwagen 1990). Similarly for
some events, such as Dante's writing of Inferno versus the
sum of Sebastian's stroll in Bologna and Caesar's crossing of Rubicon
(see Thomson 1977: 53f). More generally, intuition and common sense
suggest that some mereological composites exist, not all; yet
the question of which composites exist seems to be up for
grabs. Consider a series of almost identical mereological aggregates
that begins with a case where composition appears to obtain (e.g., the
sum of all body cells that currently make up my body, the relative
distance among any two neighboring ones being less than 1 nanometer)
and ends in a case where composition would seem not to obtain (e.g.,
the sum of all body cells that currently make up my body, after their
relative distance has been increased to 1 kilometer). Where
should we draw the line? In other words—and to limit ourselves
to (P.15ψ,i)—what value of n would mark a
change of truth-value in the soritical sequence generated by the
schema

(62)

The set of all φ-ers has a sumi if and
only if every φ is ψ,

when ‘φ’ picks out my body cells and
‘ψ’ expresses the condition ‘less than
n+1 nanometers apart from another φ-er’? It may well
be that whenever some entities compose a bigger one, it is just a
brute fact that they do so (Markosian 1998b), perhaps a matter of
contingent fact (Nolan 2005: 36, Cameron 2007). But if we are unhappy
with brute facts, if we are looking for a principled way of drawing
the line so as to specify the circumstances under which the facts
obtain, then the question is truly challenging. That is, it is a
challenging question short of treating it as a mere verbal dispute, if
not denying that it makes any sense to raise it in the first place
(see Hirsch 2005 and Putnam 1987: 16ff, respectively; see also Dorr
2005 and McGrath 2008 for relevant discussion). This is, effectively,
van Inwagen's “Special Composition Question” mentioned in
Section 4.1, an early formulation of which may be found in Hestevold
(1981). For the most part, the literature that followed has focused on
the conditions of composition for material objects, as in Sanford
(1993), Horgan (1993), Hoffman and Rosenkrantz (1997), Merricks
(2001), Hawley (2006), Markosian (2008), Vander Laan (2010), and Silva
(2013). Occasionally the question has been discussed in relation to
the ontology of actions, as in Chant (2006). In its most general form,
however, the Special Composition Question may be asked with respect to
any domain of entities whatsoever.

Concerning the second worry, to the effect that
the unrestricted sum principles in
(P.15i) would go too far, its earliest
formulations are almost as old as the principles themselves (see
e.g. V. Lowe 1953 and Rescher 1955 on the calculus of individuals,
with replies in Goodman 1956, 1958). Here one popular line of
response, inspired by Quine (1981: 10), is simply to insist that the
pattern in (P.15i) is the only plausible option,
disturbing as this might sound. Granted, common sense and intuition
dictate that some and only some mereological composites exist, but we
have just seen that it is hard to draw a principled line. On pain of
accepting brute facts, it would appear that any attempt to do away
with queer sums by restricting composition would have to do away with
too much else besides the queer entities; for queerness comes in
degrees whereas parthood and existence cannot be a matter of degree
(though we shall return to this issue in Section 5). As Lewis (1986b:
213) puts it, no restriction on composition can be vague, but unless
it is vague, it cannot fit the intuitive
desiderata. Thus, no restriction on composition could serve
the intuitions that motivate it; any restriction would be arbitrary,
hence gratuitous. And if that is the case, then either mereological
composition never obtains or else the only non-arbitrary, non-brutal
answer to the question, Under what conditions does a set have a
sumi?, would be the radical one afforded by
(P.15i): Under any condition whatsoever. (This
line of reasoning is further elaborated in Lewis 1991: 79ff as well as
in Heller 1990: 49f, Jubien 1993: 83ff; Sider 2001: 121ff, Hudson
2001: 99ff, and Van Cleve 2008: §3; for reservations and critical
discussion, see Merricks 2005, D. Smith 2006, Nolan 2006, Korman 2008,
2010, Wake 2011, Carmichael 2011, and Effingham 2009, 2011a, 2011c.)
Besides, it might be observed that any complaints about the
counterintuitiveness of unrestricted composition rest on psychological
biases that should have no bearing on the question of how the world is
actually structured. Granted, we may feel uneasy about treating
shoe-umbrellas and trout-turkeys as bona fide entities, but
that is no ground for doing away with them altogether. We may ignore
such entities when we tally up the things we care about in ordinary
contexts, but that is not to say they do not
exist. Even if one came up “with a formula that jibed
with all ordinary judgments about what counts as a unit and what does
not” (Van Cleve 1986: 145), what would that show? The
psychological factors that guide our judgments of unity simply do not
have the sort of ontological significance that should be guiding our
construction of a good mereological theory, short of thinking that
composition itself is merely a secondary quality (as in Kriegel
2008). In the words of Thomson (1998: 167): reality is like “an
over-crowded attic”, with some interesting contents and a lot of
junk, in the ordinary sense of the term. We can ignore the junk and
leave it to gather dust; but it is there and it won't go away. (One
residual problem, that such observations do not quite address,
concerns the status of cross-categorial sums. Absent any
restriction, a pluralist ontology might involve trout-turkeys and
shoe-umbrellas along with trout-promenades, shoe-virtues,
color-numbers, and what not. It is certainly possible to conceive of
some such things, as in the theory of structured propositions
mentioned in Section 2.1, or in certain neo-Aristotelian metaphysics
that construe objects as mereological sums of a “material”
and a “formal” part; see e.g. Fine 1999, 2010, Koslicki
2007, 2008, and Toner 2012. There are also theories that allow for composite objects
consisting of both “positive” and “negative”
parts, e.g., a donut, as in Hoffman and Richards 1985. At the limit,
however, the universal entity U would involve parts of
all ontological kinds. And there would seem to be nothing
arbitrary, let alone any psychological biases, in the thought that at
least such monsters should be banned. For a statement of this
view, see Simons 2003, 2006; for a reply, see Varzi 2006b.)

A third worry, which applies to all (restricted or unrestricted)
composition principles, is this. Mereology is supposed to be
ontologically “neutral”. But it is a fact that the models
of a theory cum composition principles tend to be more
densely populated than those of the corresponding composition-free
theories. If the ontological commitment of a theory is measured in
Quinean terms—via the dictum “to be is to be a
value of a bound variable” (1939: 708)—it follows that
such theories involve greater ontological commitments than their
composition-free counterparts. This is particularly worrying in the
absence of the Strong Supplementation postulate (P.5)—hence the
extensionality principle (27)—for then the ontological
exuberance of such theories may yield massive multiplication. But the
worry is a general one: composition, whether restricted or
unrestricted, is not an ontologically “innocent”
operation.

There are two lines of response to this worry (whose earliest
formulations go as far back as V. Lowe 1953). First, it could be
observed that the ontological exuberance associated with the relevant
composition principles is not substantive—that the increase of
entities in the domain of a mereological theory cum
composition principles involves no substantive additional commitments
besides those already involved in the underlying theory
without composition. This is obvious in the case of modest
principles in the spirit of (P.11ξ) and
(P.14ψ), to the effect that all suitably related
entities must have an upper bound. After all, there are small things
and there are large things, and to say that we can always find a large
thing encompassing any given small things of the right sort is not to
say much. But the same could be said with respect to those stronger
principles that require the large thing to be composed
exactly of the small things—to be their mereological
sum in some sense or other. At least, this seems reasonable in the
presence of extensionality. For in that case it can be argued that
even a sum is, in an important sense, nothing over and above
its constituent parts. The sum is just the parts “taken
together” (Baxter 1998a: 193); it is the parts “counted
loosely” (Baxter 1988b: 580); it is, effectively, “the
same portion of Reality” (Lewis 1991: 81), which is strictly a
multitude and loosely a single thing. That's why, if you proceed with
a six-pack of beer to the six-items-or-fewer checkout line at the
grocery store, the cashier is not supposed to protest your use of the
line on the ground that you have seven items: either s/he'll count the
six bottles, or s/he'll count the one pack. This thesis, known in the
literature as “composition as identity”, is by no means
uncontroversial and admits of different formulations (see van Inwagen
1994, Yi 1999, Merricks 1999, McDaniel 2008, Berto and Carrara 2009,
Carrara and Martino 2011, Cameron 2012, Wallace 2013, Cotnoir 2013a,
Hawley 2013, and the essays in Baxter and Cotnoir 2014). To the extent
that the thesis is accepted, however, the charge of ontological
exuberance loses its force. The additional entities postulated by the
sum axioms would not be a genuine addition to being; they would be, in
Armstrong's phrase, an “ontological free lunch” (1997:
13). In fact, if composition is in some sense a form of identity, then
the charge of ontological extravagance discussed in connection with
unrestricted composition loses its force, too. For if a sum is nothing
over and above its constituent proper parts, whatever they are, and if
the latter are all right, then there is nothing extravagant in
countenancing the former: it just is them, whatever they are. (This is
not to say that unrestricted composition is entailed by the
thesis that composition is identity; indeed, see McDaniel 2010 for an
argument to the effect that it isn't.)

Secondly, it could be observed that the objection in question bites at
the wrong level. If, given some entities, positing their sum were to
count as further ontological commitment, then, given a mereologically
composite entity, positing its proper parts should also count as
further commitment. After all, every entity is distinct from its
proper parts. But then the worry has nothing to do with the
composition axioms; it is, rather, a question of whether there is any
point in countenancing a whole along with its proper parts or
vice versa (see Varzi 2000 and references therein). And if the answer
is in the negative, then there seems to be little use for mereology
tout court. From the point of view of the present worry, it
would appear that the only thoroughly parsimonious account would be
one that rejects any mereological complex
whatsoever. Philosophically such an account is defensible (Rosen and
Dorr 2002; Grupp 2006; Liggins 2008; Cameron 2010; Sider 2013; Contessa 2014) and the
corresponding axiom,

(P.20)

Simplicity
Ax,

is certainly compatible with M (up to EM
and more). But the immediate corollary

(63)

Pxy ↔ x=y

says it all: nothing would be part of anything else and parthood
would collapse to identity. (This account is sometimes referred to as
mereological nihilism, in contrast to the
mereological universalism expressed by (P.15i);
see van Inwagen 1990:
72ff.[25]
Van Inwagen himself endorses a restricted version of nihilism, which
leaves room for composite living things. So does Merricks 2000, 2001,
whose restricted nihilism leaves room for composite conscious
things.)

In recent years, further worries have been raised concerning
mereological theories with substantive composition
principles—especially concerning the full strength of
GEM. Among other things, it has been argued that the
principle of unrestricted composition does not sit well with certain
fundamental intuitions about persistence through time (van Inwagen
1990, 75ff), that it is incompatible with certain plausible theories
of space (Forrest 1996b), or that it leads to paradoxes similar to the
ones afflicting naïve set theory (Bigelow 1996). A detailed
examination of such arguments is beyond the scope of this entry. For
some discussion of the first issue, however, see Rea (1998), McGrath
(1998, 2001), Hudson (2001: 93ff) and Eklund (2002: §7). On the
second, see Oppy (1997) and Mormann (1999). Hudson (2001: 95ff) also
contains some discussion of the last point.

We conclude with some remarks on a question that was briefly mentioned
above in connection with the Special Composition Question but that
pertains more generally to the underlying notion of parthood that
mereology seeks to systematize. All the theories examined so far, from
M to GEM and its variants, appear to
assume that parthood is a perfectly determinate relation: given any
two entities x and y, there is always an objective,
determinate fact of the matter as to whether or not x is part
of y. However, in some cases this seems problematic. Perhaps
there is no room for indeterminacy in the idealized mereology of space
and time as such; but when it comes to the mereology of ordinary
spatio-temporal particulars (for instance) the picture looks
different. Think of objects such as clouds, forests, heaps of
sand. What exactly are their constitutive parts? What are the
mereological boundaries of a desert, a river, a mountain? Some stuff
is positively part of Mount Everest and some stuff is positively not
part of it, but there is borderline stuff whose mereological
relationship to Everest seems indeterminate. Even living organisms
may, on closer look, give rise to indeterminacy issues. Surely
Tibbles's body comprises his tail and surely it does not comprise
Pluto's. But what about the whisker that is coming loose? It used to
be a firm part of Tibbles and soon it will drop off for good, yet
meanwhile its mereological relation to the cat is dubious. And what
goes for material bodies goes for everything. What are the
mereological boundaries of a neighborhood, a college, a social
organization? What about the boundaries of events such as promenades,
concerts, wars? What about the extensions of such ordinary concepts as
baldness, wisdom, personhood?

These worries are of no little import, and it might be thought that
some of the principles discussed above would have to be revisited
accordingly—not because of their ontological import but because
of their classical, bivalent presuppositions. For example, the
extensionality theorem of EM, (27), says that
composite things with the same proper parts are identical, but in the
presence of indeterminacy this may call for qualifications. The model
in Figure 9, left, depicts x and y as non-identical
by virtue of their having distinct determinate parts; yet one might
prefer to describe a situation of this sort as one in which the
identity between x and y is itself indeterminate,
owing to the partly indeterminate status of the two outer
atoms. Conversely, in the model on the right x and y
have the same determinate proper parts, yet again one might prefer to
suspend judgment concerning their identity, owing to the indeterminate
status of the middle atom.

Figure 9. Objects with indeterminate parts (dashed lines).

Now, it is clear that a lot here depends on how exactly one
understands the relevant notion of indeterminacy. There are, in fact,
two ways of understanding a claim of the form

(64)

It is indeterminate whether a is part of b,

depending on whether the phrase ‘it is indeterminate
whether’ is assigned wide scope, as in (64a), or narrow scope,
as in (64b):

(64a)

It is indeterminate whether b is such that a is part of it.

(64b)

b is such that it is indeterminate whether a is part of it.

On the first understanding, the indeterminacy is merely de
dicto: perhaps ‘a’ or
‘b’ are vague terms, or perhaps
‘part’ is a vague predicate, but there is no reason to
suppose that such vagueness is due to objective deficiencies in the
underlying reality. If so, then there is no reason to think that it
should affect the apparatus of mereology either, at least insofar as
the theory is meant to capture some structural features of the world
regardless of how we talk about it. For example, the statement

(65)

The loose whisker is part of Tibbles

may owe its indeterminacy to the semantic indeterminacy of
‘Tibbles’: our linguistic practices do not, on closer
look, specify exactly which portion of reality is currently picked out
by that name. In particular, they do not specify whether the name
picks out something whose current parts include the whisker that is
coming loose and, as a consequence, the truth conditions of (65) are
not fully determined. But this is not to say that the stuff out there
is mereologically indeterminate. Each one of a large variety of
slightly distinct chunks of reality has an equal claim to being the
referent of the vaguely introduced name ‘Tibbles’, and
each such thing has a perfectly precise mereological structure: some
of them currently include the lose whisker among their parts, others
do not. (Proponents of this view, which also affords a way of dealing
with the so-called “problem of the many” of Unger 1980 and
Geach 1980, include Hughes 1986, Heller 1990, Lewis 1993a, McGee 1997,
and Varzi 2001.) Alternatively, one could hold that the indeterminacy
of (65) is due, not to the semantic indeterminacy of
‘Tibbles’, but to that of ‘part’ (as in
Donnelly 2014): there is no one parthood relation; rather, several
slightly different relations are equally eligible as extension of the
parthood predicate, and while some such relations connect the loose
wisker to Tibbles, others do not. In this sense, the dashed lines in
Figure 9 would be
“defects” in the models, not in the reality that they are
meant to represent. Either way, it is apparent that, on a de dicto understanding,
mereological indeterminacy need not be due to the way the world is (or
isn't): it may just be an instance of a more general and widespread
phenomenon of indeterminacy that affects our language and our
conceptual apparatus at large. As such, it can be accounted for in
terms of whatever theory—semantic, pragmatic, or even
epistemic—one finds best suited for dealing with the phenomenon
in its generality. (See the entry on
vagueness.)
The principles of
mereology, understood as a theory of the parthood relation,
or of all the relations that qualify as admissible interpretations
of the parthood predicate, would hold
regardless.[26]

By contrast, on the second way of understanding claims of the form
(64), corresponding to (64b), the relevant indeterminacy is genuinely
de re: there is no objective fact of the matter as to whether
a is part of b, regardless of the words we use to
describe the situation. For example, on this view (65) would be
indeterminate, not because of the vagueness of ‘Tibbles’,
but because of the vagueness of Tibbles itself: there simply would be
no fact of the matter as to whether the whisker that is coming loose
is part of the cat. Similarly, the dashed lines in Figure 9 would not
reflect a “defect” in the models but a genuine, objective
deficiency in the mereological organization of the underlying
reality. As it turns out, this is not a popular view: already Russell
(1923) argued that the very idea of worldly indeterminacy betrays a
“fallacy of verbalism”, and some have gone as far as
saying that de re indeterminacy is simply not
“intelligible” (Dummett 1975: 314; Lewis 1986b: 212) or
ruled out a priori (Jackson 2001: 657). Nonetheless, several
philosophers feel otherwise and the idea that the world may include
vague entities relative to which the parthood relation is not fully
determined has received considerable attention in recent literature,
from Johnsen (1989), Tye (1990), and van Inwagen (1990: ch. 17) to
Morreau (2002), McKinnon (2003), Akiba (2004), N. Smith (2005), Hyde
(2008: §5.3), Carmichael (2011), and Sattig (2013, 2014),
inter alia. Even those who do not find that thought
attractive might wonder whether an a priori ban on it might
be unwarranted—a deep-seated metaphysical prejudice, as Burgess
(1990: 263) puts it. (Dummett himself withdrew his earlier remark and
spoke of a “prejudice” in his 1981: 440.) It is therefore
worth asking: How would such a thought impact on the mereological
theses considered in the preceding sections?

There is, unfortunately, no straightforward way of answering this
question. Broadly speaking, two main sorts of answer may be
considered, depending on whether (i) one simply takes the
indeterminacy of the parthood relation to be the reason why
certain statements involving the parthood predicate lack a definite
truth-value, or (ii) one understands the indeterminacy so that
parthood becomes a genuine matter of degree. Both options,
however, may be articulated in a variety of ways.

On option (i) (initially favored by such authors as Johnsen and Tye),
it could once again be argued that no modification of the basic
mereological machinery is strictly necessary, as long as each
postulate is taken to characterize the parthood relation insofar as
it behaves in a determinate fashion. Thus, on this approach, (P.1)
should be understood as asserting that everything is definitely part
of itself, (P.2) that any definite part of any definite part of a
thing is itself a definite part of that thing, (P.3) that things that
are definitely part of each other are identical, and so on, and the
truth of such principles is not affected by the consideration that
parthood need not be fully determinate. There is, however,
some leeway as to how such basic postulates could be integrated with
further principles concerning explicitly the indeterminate cases. For
example, do objects with indeterminate parts have indeterminate
identity? Following Evans (1978), many philosophers have taken the
answer to be obviously in the affirmative. Others, such as Cook
(1986), Sainsbury (1989), or Tye (2000), hold the opposite view: vague
objects are mereologically elusive, but they have the same precise
identity conditions as any other object. Still others maintain that
the answer depends on the strength of the underlying mereology. For
instance, T. Parsons (2000: §5.6.1) argues that on a theory such
as EMcum unrestricted binary
sums,[27]
the de re indeterminacy of (65) would be inherited by

(66)

Tibbles is identical with the sum of Tibbles and the loose whisker.

A related question is: Does countenancing objects with indeterminate
parts entail that composition be vague, i.e., that there is sometimes
no matter of fact whether some things make up a whole? A popular view,
much influenced by Lewis (1986b: 212), says that it does. Others, such
as Morreau (2002: 338), argue instead that the link between vague
parthood and vague composition is unwarranted: perhaps the de
re indeterminacy of (65) is inherited by some instances of

(67)

Tibbles is composed of x and the loose whisker.

(for example, x could be something that is just like Tibbles
except that the whisker is determinately not part of it); yet this
would not amount to saying that composition is vague, for the
following might nonetheless be true:

(68)

There is something composed of x and the loose whisker.

Finally, there is of course the general question of how one should
handle logically complex statements concerning, at least in part,
mereologically indeterminate objects. A natural choice is to rely on a
three-valued semantics of some sort, the third value being, strictly
speaking, not a truth value but rather a truth-value gap. In this
spirit, both Johnsen and Tye endorse the truth-tables of Kleene (1938)
while Hyde those of Łukasiewicz (1920). However, it is worth
stressing that other choices are available, including
non-truth-functional accounts. For example, Akiba (2000) and Morreau
(2002) recommend a form of “supervaluationism”. This was
originally put forward by Fine (1975) as a theory for dealing
with de dicto indeterminacy, the idea being that a statement
involving vague expressions should count as true (false) if and only it
is true (false) on every “precisification” of those
expressions. Still, a friend of de re indeterminacy may
exploit the same idea by speaking instead of precisifications of the
underlying reality—what Sainsbury (1989) calls
“approximants”, Cohn and Gotts (1996)
“crispings”, and T. Parsons (2000) “resolutions”
of vague objects. As a result, one would be able to explain why, for
example, (69) appears to be true and (70) false (assuming that
Tibbles's head is definitely part of Tibbles), whereas both
conditionals would be equally indeterminate on Kleene's semantics and
equally true on Łukasiewicz's:

(69)

If the loose whisker is part of the head and the head is part of
Tibbles, then the whisker itself is part of Tibbles.

(70)

If the loose whisker is part of the head and the head is part of
Tibbles, then the whisker itself is not part of Tibbles.

As for option (ii)—to the effect that de re
mereological indeterminacy is a matter of degree—the picture is
different. Here the main motivation is that whether or not something
is part of something else is really not an all-or-nothing affair. If
Tibbles has two whiskers that are coming loose, then we may want to
say that neither is a definite part of Tibbles. But if one whisker is
looser than the other, then it would seem plausible to say that the
first is part of Tibbles to a lesser degree than the second,
and one may want the postulates of mereology to be sensitive to such
distinctions. This is, for example, van Inwagen's (1990) view of the
matter, which results in a fuzzification of parthood that parallels in
many ways to the fuzzification of membership in Zadeh's (1965) set
theory, and it is this sort of intuition that also led to the
development of such formal theories as Polkowsky and Skowron's (1994)
“rough mereology” or N. Smith's (2005) theory of
“concrete parts”. Again, there is room for some leeway
concerning matters of detail, but in this case the main features of
the approach are fairly clear and uniform across the literature. For
let π be the characteristic function associated with the parthood
relation denoted by the basic mereological primitive,
‘P’. Then, if classically this function is bivalent, which
can be expressed by saying that π(x, y) always
takes, say, the value 1 or the value 0 according to whether or
not x is part of y, to say that parthood may be
indeterminate is to say that π need not be fully bivalent. And
whereas option (i) simply takes this to mean that π may sometimes
be undefined, option (ii) can be characterized by saying that the
range of π may include values intermediate between 0 and 1, i.e.,
effectively, values from the closed real interval [0, 1]. In other
words, on this latter approach π is still a perfectly standard,
total function, and the only serious question that needs to be
addressed is the genuinely mereological question of what conditions
should be assumed to characterize its behavior—a question not
different from the one that we have considered for the bivalent case
throughout the preceding sections.

This is not to say that the question is an easy one. As it turns out,
the “fuzzification” of the core theory M
is rather straightforward, but its extensions give rise to various
issues. Thus, consider the partial ordering axioms
(P.1)–(P.3). Classically, these correspond to the following conditions
on π :

(P.1π)

π(x, x) = 1

(P.2π)

π(x, z) ≥
min(π(x, y), π(y, z))

(P.3π)

If π(x, y) = 1 and
π(y, x) = 1, then x = y,

and one could argue that the very same conditions may be taken to fix
the basic properties of parthood regardless of whether π is
bivalent. Perhaps one may consider weakening (P.2π) as
follows (Polkowsky and Skowron 1994):

(P.2π′)

If π(y, z) = 1, then
π(x, z) ≥ π(x, y).

Or one may consider strengthening (P.3π) as follows (N. Smith
2005):[28]

(P.3π′)

If π(x, y) > 0 and
π(y, x) > 0, then x
= y.

But that is about it: there is little room for further
adjustments. Things immediately get complicated, though, as soon as we
move beyond M. Take, for instance, the
Supplementation principle (P.4) of MM. One natural
way of expressing it in terms of π is as follows:

(P.4π)

If π(x, y) = 1 and x
≠ y, then π(z, y) = 1 for
some z such that, for all w, either
π(w, z) = 0 or π(w, x) =
0.

There are, however, fifteen other ways of expressing (P.4) in terms of
π, obtained by re-writing one or both occurrences of ‘=
1’ as ‘> 0’ and one or both occurrences of
‘= 0’ as ‘< 1’. In the presence of
bivalence, these would all be equivalent ways of saying the same
thing. However, such alternative formulations would not coincide if
π is allowed to take non-integral values, and the question of which
version(s) best reflect the supplementation intuition would have to be
carefully examined. (See e.g. the discussion in N. Smith 2005: 397.) And
this is just the beginning: it is clear that similar issues arise with
most other principles discussed in the previous sections, such as
Complementation, Density, or the various composition principles. (See
e.g. Polkowsky and Skowron 1994: 86 for a formulation of the
Unrestricted Sum axiom (P.152).)

On the other hand, it is worth noting that precisely because the
difficulty is mainly technical—the framework itself being fairly
firm—now some of the questions raised in connection with option
(i) tend to be less open to controversy. For example, the question of
whether mereological indeterminacy implies vague identity is generally
answered in the negative, especially if one adheres to the spirit of
extensionality. For then it is natural to say that non-atomic objects are
identical if and only if they have exactly the same parts to the same
degree—and that is not a vague matter (a point already made in
Williamson 1994: 255). In other words, given that classically the
extensionality principle (27) corresponds to the following
condition:

(27π)

If there is a z such that either
π(z, x) = 1 or π(z, y) = 1,
then x = y if and only if, for every z,
π(z, x) = π(z, y),

it seems perfectly natural to stick to this condition even if the
range of π is extended from {0, 1} to [0, 1]. Likewise, the
question of whether mereological indeterminacy implies vague
composition or vague existence is generally answered in the
affirmative (though not always; see e.g. Donnelly 2009 and Barnes and
Williams 2009). Van Inwagen (1990: 228) takes this to be a rather
obvious consequence of the approach, but N. Smith (2005: 399ff) goes
further and provides a detailed analysis of how one can calculate the
degree to which a given non-empty set of things has a sum, i.e., the
degree of existence of the sum. (Roughly, the idea is to begin with
the sum as it would exist if every element of the set were a definite
part of it, and then calculate the actual degree of existence of the
sum as a function of the degree to which each element of the set is
actually part of it).

The one question that remains widely open is how all of this should be
reflected in the semantics of our language, specifically the semantics
of logically complex statements. As a matter of fact, there is a
tendency to regard this question as part and parcel of the more
general problem of choosing the appropriate semantics for fuzzy logic,
which typically amounts to an infinitary generalization of some
truth-functional three-valued semantics. The range of possibilities,
however, is broader, and even here there is room for
non-truth-functional approaches—including degree-theoretic
variants of supervaluationism (as recommended e.g. in Sanford 1993:
225).

The SEP would like to congratulate the National Endowment for the Humanities on its 50th anniversary and express our indebtedness for the five generous grants it awarded our project from 1997 to 2007.
Readers who have benefited from the SEP are encouraged to examine the NEH’s anniversary page and, if inspired to do so, send a testimonial to neh50@neh.gov.